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DrawMorseLink

PD Presentation:

X8192 X14,4,15,3 X22,9,13,10 X20,11,21,12 X12,19,7,20 X10,21,11,22 X18,14,19,13 X16,6,17,5 X2738 X4,16,5,15 X6,18,1,17

Gauss Code:

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

Jones Polynomial:

q-5 - q-4 + 4q-3 - 5q-2 + 8q-1 - 9 + 10q - 9q2 + 8q3 - 5q4 + 3q5 - q6

A2 (sl(3)) Invariant:

q-16 + 2q-14 + 2q-12 + 4q-10 + 3q-8 + 3q-6 + 4q-4 + q-2 + 3 + q2 + 2q6 - q8 + 2q10 + q16 - q18

HOMFLY-PT Polynomial:

- 2a-4z2 - a-4z4 + a-2 + 2a-2z2 + 3a-2z4 + a-2z6 + z-2 + 2 + 2z2 + 3z4 + z6 - 2a2z-2 - 6a2 - 7a2z2 - 2a2z4 + a4z-2 + 3a4 + a4z2

Kauffman Polynomial:

a-7z3 - a-6z2 + 3a-6z4 - 3a-5z3 + 5a-5z5 + 4a-4z2 - 10a-4z4 + 7a-4z6 + 5a-3z3 - 13a-3z5 + 7a-3z7 - 2a-2 + 7a-2z2 - 5a-2z4 - 8a-2z6 + 5a-2z8 + 3a-1z3 - 4a-1z5 - 5a-1z7 + 3a-1z9 - z-2 + 3 - z2 + 8z4 - 11z6 + z8 + z10 + 2az-1 - 9az + 4az3 + 15az5 - 16az7 + 4az9 - 2a2z-2 + 11a2 - 20a2z2 + 17a2z4 - 3a2z6 - 3a2z8 + a2z10 + 2a3z-1 - 9a3z + 10a3z3 + a3z5 - 4a3z7 + a3z9 - a4z-2 + 7a4 - 17a4z2 + 17a4z4 - 7a4z6 + a4z8

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 13                       1 j = 11                     2   j = 9                   3 1   j = 7                 5 2     j = 5               5 4       j = 3             5 4         j = 1           6 7           j = -1         2 3             j = -3       3 6               j = -5     1 2                 j = -7     3                   j = -9 1 1                     j = -11 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, 518]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 518]]
Out[4]=	PD[X[8, 1, 9, 2], X[14, 4, 15, 3], X[22, 9, 13, 10], X[20, 11, 21, 12], > X[12, 19, 7, 20], X[10, 21, 11, 22], X[18, 14, 19, 13], X[16, 6, 17, 5], > X[2, 7, 3, 8], X[4, 16, 5, 15], X[6, 18, 1, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10, 8, -11}, {9, -1, 3, -6, 4, -5}, > {7, -2, 10, -8, 11, -7, 5, -4, 6, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-5 -4 4 5 8 2 3 4 5 6 -9 + q - q + -- - -- + - + 10 q - 9 q + 8 q - 5 q + 3 q - q 3 2 q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 2 2 4 3 3 4 -2 2 6 8 10 3 + q + --- + --- + --- + -- + -- + -- + q + q + 2 q - q + 2 q + 14 12 10 8 6 4 q q q q q q 16 18 > q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 518]][a, z]
Out[8]=	2 4 2 2 -2 2 4 -2 2 a a 2 2 z 2 z 2 2 2 + a - 6 a + 3 a + z - ---- + -- + 2 z - ---- + ---- - 7 a z + 2 2 4 2 z z a a 4 4 6 4 2 4 z 3 z 2 4 6 z > a z + 3 z - -- + ---- - 2 a z + z + -- 4 2 2 a a a
In[9]:=	Kauffman[Link[11, Alternating, 518]][a, z]
Out[9]=	2 4 3 2 2 4 -2 2 a a 2 a 2 a 3 2 3 - -- + 11 a + 7 a - z - ---- - -- + --- + ---- - 9 a z - 9 a z - z - 2 2 2 z z a z z 2 2 2 3 3 3 3 z 4 z 7 z 2 2 4 2 z 3 z 5 z 3 z 3 > -- + ---- + ---- - 20 a z - 17 a z + -- - ---- + ---- + ---- + 4 a z + 6 4 2 7 5 3 a a a a a a a 4 4 4 5 3 3 4 3 z 10 z 5 z 2 4 4 4 5 z > 10 a z + 8 z + ---- - ----- - ---- + 17 a z + 17 a z + ---- - 6 4 2 5 a a a a 5 5 6 6 13 z 4 z 5 3 5 6 7 z 8 z 2 6 4 6 > ----- - ---- + 15 a z + a z - 11 z + ---- - ---- - 3 a z - 7 a z + 3 a 4 2 a a a 7 7 8 9 7 z 5 z 7 3 7 8 5 z 2 8 4 8 3 z > ---- - ---- - 16 a z - 4 a z + z + ---- - 3 a z + a z + ---- + 3 a 2 a a a 9 3 9 10 2 10 > 4 a z + a z + z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 1 1 3 1 2 3 6 7 q + 5 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ----- + 11 6 9 6 9 5 7 4 5 4 5 3 3 3 3 2 q t q t q t q t q t q t q t q t 2 3 6 q 3 5 5 2 7 2 7 3 > ---- + --- + --- + 4 q t + 5 q t + 4 q t + 5 q t + 2 q t + 2 q t t q t 9 3 9 4 11 4 13 5 > 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a518

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