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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-4 + 3q-3 - 5q-2 + 9q-1 - 11 + 14q - 13q2 + 13q3 - 9q4 + 6q5 - 3q6 + q7

A2 (sl(3)) Invariant:

- q-12 + q-8 + 3q-4 + 3 + 4q2 + 2q4 + 7q6 + q8 + 4q10 + q12 + 2q16 - q18 + q20

HOMFLY-PT Polynomial:

a-4z-2 + 3a-4 + 5a-4z2 + 4a-4z4 + a-4z6 - 2a-2z-2 - 8a-2 - 16a-2z2 - 14a-2z4 - 6a-2z6 - a-2z8 + z-2 + 6 + 12z2 + 9z4 + 2z6 - a2 - 3a2z2 - a2z4

Kauffman Polynomial:

- a-8z2 + a-8z4 - 3a-7z3 + 3a-7z5 - a-6 + 3a-6z2 - 6a-6z4 + 5a-6z6 + 4a-5z3 - 8a-5z5 + 6a-5z7 - a-4z-2 + 4a-4 - 6a-4z2 + 10a-4z4 - 11a-4z6 + 6a-4z8 + 2a-3z-1 - 6a-3z + 8a-3z3 - 3a-3z5 - 5a-3z7 + 4a-3z9 - 2a-2z-2 + 12a-2 - 37a-2z2 + 56a-2z4 - 40a-2z6 + 9a-2z8 + a-2z10 + 2a-1z-1 - 8a-1z + 4a-1z3 + 14a-1z5 - 21a-1z7 + 7a-1z9 - z-2 + 10 - 36z2 + 56z4 - 37z6 + 6z8 + z10 - 3az + 7az3 + 2az5 - 9az7 + 3az9 + 2a2 - 9a2z2 + 17a2z4 - 13a2z6 + 3a2z8 - a3z + 4a3z3 - 4a3z5 + a3z7

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 15                       1 j = 13                     2   j = 11                   4 1   j = 9                 6 3     j = 7               7 3       j = 5             7 7         j = 3           7 6           j = 1         5 8             j = -1       4 6               j = -3     2 6                 j = -5   1 3                   j = -7   2                     j = -9 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, 515]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 515]]
Out[4]=	PD[X[8, 1, 9, 2], X[14, 4, 15, 3], X[20, 11, 21, 12], X[18, 10, 19, 9], > X[22, 19, 13, 20], X[10, 14, 11, 13], X[12, 21, 7, 22], 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, 4, -6, 3, -7}, > {6, -2, 10, -8, 11, -4, 5, -3, 7, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-4 3 5 9 2 3 4 5 6 7 -11 - q + -- - -- + - + 14 q - 13 q + 13 q - 9 q + 6 q - 3 q + q 3 2 q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 -8 3 2 4 6 8 10 12 16 18 20 3 - q + q + -- + 4 q + 2 q + 7 q + q + 4 q + q + 2 q - q + q 4 q
In[8]:=	HOMFLYPT[Link[11, Alternating, 515]][a, z]
Out[8]=	2 2 3 8 2 -2 1 2 2 5 z 16 z 2 2 6 + -- - -- - a + z + ----- - ----- + 12 z + ---- - ----- - 3 a z + 4 2 4 2 2 2 4 2 a a a z a z a a 4 4 6 6 8 4 4 z 14 z 2 4 6 z 6 z z > 9 z + ---- - ----- - a z + 2 z + -- - ---- - -- 4 2 4 2 2 a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 515]][a, z]
Out[9]=	-6 4 12 2 -2 1 2 2 2 6 z 8 z 10 - a + -- + -- + 2 a - z - ----- - ----- + ---- + --- - --- - --- - 4 2 4 2 2 2 3 a z 3 a a a a z a z a z a 2 2 2 2 3 3 3 2 z 3 z 6 z 37 z 2 2 3 z 4 z > 3 a z - a z - 36 z - -- + ---- - ---- - ----- - 9 a z - ---- + ---- + 8 6 4 2 7 5 a a a a a a 3 3 4 4 4 4 8 z 4 z 3 3 3 4 z 6 z 10 z 56 z > ---- + ---- + 7 a z + 4 a z + 56 z + -- - ---- + ----- + ----- + 3 a 8 6 4 2 a a a a a 5 5 5 5 6 2 4 3 z 8 z 3 z 14 z 5 3 5 6 5 z > 17 a z + ---- - ---- - ---- + ----- + 2 a z - 4 a z - 37 z + ---- - 7 5 3 a 6 a a a a 6 6 7 7 7 11 z 40 z 2 6 6 z 5 z 21 z 7 3 7 8 > ----- - ----- - 13 a z + ---- - ---- - ----- - 9 a z + a z + 6 z + 4 2 5 3 a a a a a 8 8 9 9 10 6 z 9 z 2 8 4 z 7 z 9 10 z > ---- + ---- + 3 a z + ---- + ---- + 3 a z + z + --- 4 2 3 a 2 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 2 1 3 2 6 4 6 5 q 8 q + 7 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + --- + 9 5 7 4 5 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t q t q t 3 5 5 2 7 2 7 3 9 3 9 4 > 6 q t + 7 q t + 7 q t + 7 q t + 3 q t + 6 q t + 3 q t + 11 4 11 5 13 5 15 6 > 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a515

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