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DrawMorseLink

PD Presentation:

X6172 X14,4,15,3 X16,5,17,6 X20,9,21,10 X18,8,19,7 X22,19,13,20 X8,14,9,13 X12,15,5,16 X10,21,11,22 X2,11,3,12 X4,18,1,17

Gauss Code:

{{1, -10, 2, -11}, {3, -1, 5, -7, 4, -9, 10, -8}, {7, -2, 8, -3, 11, -5, 6, -4, 9, -6}}

Jones Polynomial:

- q-8 + 5q-7 - 11q-6 + 17q-5 - 22q-4 + 26q-3 - 24q-2 + 22q-1 - 14 + 9q - 4q2 + q3

A2 (sl(3)) Invariant:

- q-24 + 2q-22 - 2q-18 + 4q-16 - 4q-14 + 3q-12 + 3q-10 + 2q-8 + 10q-6 + 8q-2 + 1 - q2 + 3q4 - 2q6 + q8

HOMFLY-PT Polynomial:

z-2 + 2 + 3z2 + 3z4 + z6 - 2a2z-2 - 3a2 - 8a2z2 - 10a2z4 - 5a2z6 - a2z8 + a4z-2 + a4 + 5a4z2 + 6a4z4 + 2a4z6 - a6z2 - a6z4

Kauffman Polynomial:

a-2z2 - 2a-2z4 + a-2z6 + 5a-1z3 - 9a-1z5 + 4a-1z7 - z-2 + 3 - 6z2 + 13z4 - 16z6 + 7z8 + 2az-1 - 3az + 8az3 - 8az5 - 5az7 + 6az9 - 2a2z-2 + 5a2 - 20a2z2 + 45a2z4 - 47a2z6 + 16a2z8 + 2a2z10 + 2a3z-1 - 3a3z + 12a3z3 - 7a3z5 - 17a3z7 + 14a3z9 - a4z-2 + 3a4 - 14a4z2 + 40a4z4 - 53a4z6 + 22a4z8 + 2a4z10 + 13a5z3 - 24a5z5 + 3a5z7 + 8a5z9 - a6z2 + 6a6z4 - 18a6z6 + 13a6z8 + 4a7z3 - 15a7z5 + 11a7z7 - 4a8z4 + 5a8z6 + a9z5

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 7                       1 j = 5                     3   j = 3                   6 1   j = 1                 8 3     j = -1               14 6       j = -3             13 11         j = -5           13 11           j = -7         9 13             j = -9       8 13               j = -11     4 10                 j = -13   1 7                   j = -15   4                     j = -17 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Length[Skeleton[L]]
Out[2]=	3
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 458]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 458]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 4, 15, 3], X[16, 5, 17, 6], X[20, 9, 21, 10], > X[18, 8, 19, 7], X[22, 19, 13, 20], X[8, 14, 9, 13], X[12, 15, 5, 16], > X[10, 21, 11, 22], X[2, 11, 3, 12], X[4, 18, 1, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {3, -1, 5, -7, 4, -9, 10, -8}, > {7, -2, 8, -3, 11, -5, 6, -4, 9, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-8 5 11 17 22 26 24 22 2 3 -14 - q + -- - -- + -- - -- + -- - -- + -- + 9 q - 4 q + q 7 6 5 4 3 2 q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 2 2 4 4 3 3 2 10 8 2 4 1 - q + --- - --- + --- - --- + --- + --- + -- + -- + -- - q + 3 q - 22 18 16 14 12 10 8 6 2 q q q q q q q q q 6 8 > 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 458]][a, z]
Out[8]=	2 4 2 4 -2 2 a a 2 2 2 4 2 6 2 4 2 - 3 a + a + z - ---- + -- + 3 z - 8 a z + 5 a z - a z + 3 z - 2 2 z z 2 4 4 4 6 4 6 2 6 4 6 2 8 > 10 a z + 6 a z - a z + z - 5 a z + 2 a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 458]][a, z]
Out[9]=	2 4 3 2 2 4 -2 2 a a 2 a 2 a 3 2 z 3 + 5 a + 3 a - z - ---- - -- + --- + ---- - 3 a z - 3 a z - 6 z + -- - 2 2 z z 2 z z a 3 2 2 4 2 6 2 5 z 3 3 3 5 3 > 20 a z - 14 a z - a z + ---- + 8 a z + 12 a z + 13 a z + a 4 5 7 3 4 2 z 2 4 4 4 6 4 8 4 9 z > 4 a z + 13 z - ---- + 45 a z + 40 a z + 6 a z - 4 a z - ---- - 2 a a 6 5 3 5 5 5 7 5 9 5 6 z 2 6 > 8 a z - 7 a z - 24 a z - 15 a z + a z - 16 z + -- - 47 a z - 2 a 7 4 6 6 6 8 6 4 z 7 3 7 5 7 > 53 a z - 18 a z + 5 a z + ---- - 5 a z - 17 a z + 3 a z + a 7 7 8 2 8 4 8 6 8 9 3 9 > 11 a z + 7 z + 16 a z + 22 a z + 13 a z + 6 a z + 14 a z + 5 9 2 10 4 10 > 8 a z + 2 a z + 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	11 14 1 4 1 7 4 10 8 13 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 9 3 q q t q t q t q t q t q t q t q t 9 13 13 11 13 6 t 2 3 2 > ----- + ----- + ----- + ---- + ---- + --- + 8 q t + 3 q t + 6 q t + 7 3 7 2 5 2 5 3 q q t q t q t q t q t 3 3 5 3 7 4 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a458

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