Link Splice Base

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[[Image:Data:Link Splice Base/Previous Knot.gif|80px|link=Data:Link Splice Base/Previous Knot]] [[Data:Link Splice Base/Previous Knot]]

[[Image:Data:Link Splice Base/Next Knot.gif|80px|link=Data:Link Splice Base/Next Knot]] [[Data:Link Splice Base/Next Knot]]

File:Link Splice Base.gif

Visit [<*KnotilusURL[K]*> Link Splice Base's page] at Knotilus! Visit <*n*><*If [AlternatingQ[K,"a","n"]*><*k*>.html Link Splice Base's page] at the original Knot Atlas!

Link Splice Base Quick Notes

Link Splice Base Further Notes and Views

Knot presentations

Planar diagram presentation	Data:Link Splice Base/PD Presentation
Gauss code	Data:Link Splice Base/Gauss Code

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	Data:Link Splice Base/Multivariable Alexander (db)
Jones polynomial	Data:Link Splice Base/Jones Polynomial (db)
Signature	Data:Link Splice Base/Signature (db)
HOMFLY-PT polynomial	Data:Link Splice Base/HOMFLYPT Polynomial (db)
Kauffman polynomial	Data:Link Splice Base/Kauffman Polynomial (db)

Vassiliev invariants

V2 and V3:

(Data:Link Splice Base/V 2, Data:Link Splice Base/V 3)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 Data:Link Splice Base/V 2,1 Data:Link Splice Base/V 3,1 Data:Link Splice Base/V 4,1 Data:Link Splice Base/V 4,2 Data:Link Splice Base/V 4,3 Data:Link Splice Base/V 5,1 Data:Link Splice Base/V 5,2 Data:Link Splice Base/V 5,3 Data:Link Splice Base/V 5,4 Data:Link Splice Base/V 6,1 Data:Link Splice Base/V 6,2 Data:Link Splice Base/V 6,3 Data:Link Splice Base/V 6,4 Data:Link Splice Base/V 6,5 Data:Link Splice Base/V 6,6 Data:Link Splice Base/V 6,7 Data:Link Splice Base/V 6,8 Data:Link Splice Base/V 6,9

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s+1$, where $s=$Data:Link Splice Base/Signature is the signature of Link Splice Base. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

<*TabularKh[Kh[K][q, t], KnotSignature[K]+{1,-1}]*>

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

<*InOut["Crossings[``]", K]*> <*InOut["PD[``]", K]*> <*InOut["GaussCode[``]", K]*> <*InOut["BR[``]", K]*> <*InOut["alex = Alexander[``][t]", K]*> <*InOut["Conway[``][z]", K]*> <*InOut["Select[AllKnots[], (alex === Alexander[#][t])&]"]*> <*InOut["{KnotDet[`1`], KnotSignature[`1`]}", K]*> <*InOut["J=Jones[``][q]", K]*> <*InOut[

 "Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]"

]*> <* If[Crossings[K]<=18, Include["ColouredJonesM.mhtml"] ,""] *> <*InOut["A2Invariant[``][q]", K]*> <*InOut["Kauffman[``][a, z]", K]*> <*InOut["{Vassiliev[2][`1`], Vassiliev[3][`1`]}", K ]*> <*InOut["Kh[``][q, t]", K]*>

In[1]:=	<< KnotTheory`
<*InOut[1]; KnotTheoryWelcomeMessage[]*>

<* (* *) *>