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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-2 + 5q-1 - 10 + 16q - 20q2 + 23q3 - 22q4 + 19q5 - 12q6 + 8q7 - 3q8 + q9

A2 (sl(3)) Invariant:

- q-6 + 3q-4 - q-2 - 1 + 5q2 - 4q4 + 3q6 - q10 + 4q12 - q14 + 8q16 + 4q18 + 2q20 + 6q22 + q28

HOMFLY-PT Polynomial:

a-8z-2 + a-8 + a-8z2 - 2a-6z-2 - 3a-6 - 3a-6z2 - 2a-6z4 + a-4z-2 + a-4 + a-4z2 + a-4z4 + a-4z6 + a-2z4 + a-2z6 + 1 - z4

Kauffman Polynomial:

2a-10z2 - 3a-10z4 + a-10z6 + 3a-9z3 - 7a-9z5 + 3a-9z7 - a-8z-2 + 8a-8 - 20a-8z2 + 24a-8z4 - 18a-8z6 + 6a-8z8 + 2a-7z-1 - 11a-7z + 17a-7z3 - 7a-7z5 - 6a-7z7 + 5a-7z9 - 2a-6z-2 + 13a-6 - 31a-6z2 + 46a-6z4 - 36a-6z6 + 10a-6z8 + 2a-6z10 + 2a-5z-1 - 11a-5z + 17a-5z3 + a-5z5 - 19a-5z7 + 12a-5z9 - a-4z-2 + 5a-4 - 10a-4z2 + 26a-4z4 - 36a-4z6 + 15a-4z8 + 2a-4z10 + 6a-3z3 - 14a-3z5 + 7a-3z9 - a-2z2 + 2a-2z4 - 14a-2z6 + 11a-2z8 + 3a-1z3 - 14a-1z5 + 10a-1z7 + 1 - 5z4 + 5z6 + az5

Khovanov Homology:

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 19                       1 j = 17                     3 1 j = 15                   5     j = 13                 7 3     j = 11               12 5       j = 9             11 8         j = 7           12 11           j = 5         8 11             j = 3       8 12               j = 1     4 10                 j = -1   1 6                   j = -3   4                     j = -5 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, 460]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 460]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 4, 15, 3], X[22, 10, 13, 9], X[20, 8, 21, 7], > X[8, 14, 9, 13], X[18, 15, 19, 16], X[16, 6, 17, 5], X[12, 18, 5, 17], > X[10, 22, 11, 21], X[2, 11, 3, 12], X[4, 20, 1, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {7, -1, 4, -5, 3, -9, 10, -8}, > {5, -2, 6, -7, 8, -6, 11, -4, 9, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-2 5 2 3 4 5 6 7 8 9 -10 - q + - + 16 q - 20 q + 23 q - 22 q + 19 q - 12 q + 8 q - 3 q + q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-6 3 -2 2 4 6 10 12 14 16 18 -1 - q + -- - q + 5 q - 4 q + 3 q - q + 4 q - q + 8 q + 4 q + 4 q 20 22 28 > 2 q + 6 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 460]][a, z]
Out[8]=	2 2 2 4 4 -8 3 -4 1 2 1 z 3 z z 4 2 z z 1 + a - -- + a + ----- - ----- + ----- + -- - ---- + -- - z - ---- + -- + 6 8 2 6 2 4 2 8 6 4 6 4 a a z a z a z a a a a a 4 6 6 z z z > -- + -- + -- 2 4 2 a a a
In[9]:=	Kauffman[Link[11, Alternating, 460]][a, z]
Out[9]=	2 8 13 5 1 2 1 2 2 11 z 11 z 2 z 1 + -- + -- + -- - ----- - ----- - ----- + ---- + ---- - ---- - ---- + ---- - 8 6 4 8 2 6 2 4 2 7 5 7 5 10 a a a a z a z a z a z a z a a a 2 2 2 2 3 3 3 3 3 20 z 31 z 10 z z 3 z 17 z 17 z 6 z 3 z 4 > ----- - ----- - ----- - -- + ---- + ----- + ----- + ---- + ---- - 5 z - 8 6 4 2 9 7 5 3 a a a a a a a a a 4 4 4 4 4 5 5 5 5 5 3 z 24 z 46 z 26 z 2 z 7 z 7 z z 14 z 14 z > ---- + ----- + ----- + ----- + ---- - ---- - ---- + -- - ----- - ----- + 10 8 6 4 2 9 7 5 3 a a a a a a a a a a 6 6 6 6 6 7 7 7 5 6 z 18 z 36 z 36 z 14 z 3 z 6 z 19 z > a z + 5 z + --- - ----- - ----- - ----- - ----- + ---- - ---- - ----- + 10 8 6 4 2 9 7 5 a a a a a a a a 7 8 8 8 8 9 9 9 10 10 10 z 6 z 10 z 15 z 11 z 5 z 12 z 7 z 2 z 2 z > ----- + ---- + ----- + ----- + ----- + ---- + ----- + ---- + ----- + ----- a 8 6 4 2 7 5 3 6 4 a a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 4 1 6 4 q 3 5 5 2 10 q + 8 q + ----- + ----- + ---- + --- + --- + 12 q t + 8 q t + 11 q t + 5 3 3 2 2 q t t q t q t q t 7 2 7 3 9 3 9 4 11 4 11 5 > 12 q t + 11 q t + 11 q t + 8 q t + 12 q t + 5 q t + 13 5 13 6 15 6 17 7 17 8 19 8 > 7 q t + 3 q t + 5 q t + 3 q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a460

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