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Knot Signature


The signature s(K) of a knot K can be defined using the skein relationship

 s(unknot)=0
(1)
 s(K_+)-s(K_-) in {0,2},
(2)

and

 4|s(K)<->del (K)(2i)>0,
(3)

where del (K) is the Conway polynomial and del (K)(2i) is an odd number.

Many unknotting numbers can be determined using a knot's signature.

Knot signatures are implemented in the Wolfram Language as KnotData[knot, "Signature"]. The following table summarizes knot signatures for knots on 10 of fewer crossings.

0_108_(16)29_(25)-210_6-410_(36)-210_(66)-610_(96)010_(126)210_(156)2
3_1-28_(17)09_(26)210_7-210_(37)010_(67)-210_(97)-210_(127)-410_(157)4
4_108_(18)09_(27)010_8-410_(38)-210_(68)010_(98)-410_(128)-610_(158)0
5_1-48_(19)-69_(28)-210_9-210_(39)-410_(69)210_(99)010_(129)010_(159)-2
5_2-28_(20)09_(29)210_(10)010_(40)210_(70)210_(100)410_(130)010_(160)-4
6_108_(21)-29_(30)010_(11)-210_(41)-210_(71)010_(101)-410_(131)-210_(161)-4
6_2-29_1-89_(31)-210_(12)210_(42)010_(72)-410_(102)010_(132)010_(162)2
6_309_2-29_(32)210_(13)010_(43)010_(73)210_(103)210_(133)-210_(163)-2
7_1-69_3-69_(33)010_(14)-410_(44)-210_(74)-210_(104)010_(134)-610_(164)0
7_2-29_4-49_(34)010_(15)210_(45)010_(75)010_(105)-210_(135)010_(165)2
7_3-49_5-29_(35)-210_(16)-210_(46)-610_(76)-410_(106)-210_(136)2
7_4-29_6-69_(36)410_(17)010_(47)410_(77)210_(107)010_(137)0
7_5-49_7-49_(37)010_(18)-210_(48)010_(78)-410_(108)-210_(138)-2
7_6-29_8-29_(38)-410_(19)-210_(49)-610_(79)010_(109)010_(139)-6
7_709_9-69_(39)210_(20)-210_(50)-410_(80)-610_(110)-210_(140)0
8_109_(10)-49_(40)-210_(21)-410_(51)210_(81)010_(111)-410_(141)0
8_2-49_(11)49_(41)010_(22)010_(52)-210_(82)-210_(112)210_(142)-6
8_309_(12)-29_(42)210_(23)210_(53)-410_(83)210_(113)-210_(143)2
8_429_(13)-49_(43)-410_(24)-210_(54)210_(84)-210_(114)010_(144)-2
8_5-49_(14)09_(44)010_(25)-410_(55)-410_(85)410_(115)010_(145)2
8_6-29_(15)29_(45)210_(26)010_(56)-410_(86)010_(116)210_(146)0
8_729_(16)-69_(46)010_(27)210_(57)210_(87)010_(117)210_(147)-2
8_809_(17)-29_(47)-210_(28)010_(58)010_(88)010_(118)010_(148)2
8_909_(18)-49_(48)210_(29)-210_(59)-210_(89)210_(119)010_(149)-4
8_(10)29_(19)09_(49)-410_(30)-210_(60)010_(90)010_(120)-410_(150)-4
8_(11)-29_(20)-410_1010_(31)010_(61)-410_(91)010_(121)-210_(151)2
8_(12)09_(21)210_2-610_(32)010_(62)410_(92)-410_(122)010_(152)-6
8_(13)09_(22)-210_3010_(33)010_(63)-410_(93)210_(123)010_(153)0
8_(14)-29_(23)-410_4210_(34)010_(64)-210_(94)-210_(124)-810_(154)-4
8_(15)-49_(24)010_5410_(35)010_(65)210_(95)210_(125)210_(155)0

See also

Skein Relationship, Unknotting Number

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References

Gordon, C. M.; Litherland, R. A.; and Murasugi, K. "Signatures of Covering Links." Canad. J. Math. 33, 381-394, 1981.Murasugi, K. "On the Signature of Links." Topology 9, 283-298, 1970.Murasugi, K. "Signatures and Alexander Polynomials of Two-Bridge Knots." C. R. Math. Rep. Acad. Sci. Canada 5, 133-136, 1983.Murasugi, K. "On the Signature of a Graph." C. R. Math. Rep. Acad. Sci. Canada 10, 107-111, 1988.Murasugi, K. "On Invariants of Graphs with Applications to Knot Theory." Trans. Amer. Math. Soc. 314, 1-49, 1989.Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, 1976.Stoimenow, A. "Signatures." http://www.ms.u-tokyo.ac.jp/~stoimeno/ptab/sig10.html.

Referenced on Wolfram|Alpha

Knot Signature

Cite this as:

Weisstein, Eric W. "Knot Signature." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/KnotSignature.html

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