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DrawMorseLink

PD Presentation:

X8192 X2,9,3,10 X10,3,11,4 X16,8,17,7 X22,16,7,15 X6,21,1,22 X18,11,19,12 X14,6,15,5 X20,13,21,14 X12,19,13,20 X4,18,5,17

Gauss Code:

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

Jones Polynomial:

q-17/2 - 3q-15/2 + 6q-13/2 - 10q-11/2 + 13q-9/2 - 15q-7/2 + 14q-5/2 - 13q-3/2 + 9q-1/2 - 6q1/2 + 3q3/2 - q5/2

A2 (sl(3)) Invariant:

- q-24 + q-22 - 2q-20 + 2q-18 + 3q-12 - 2q-10 + 5q-8 - q-6 + 2q-4 + q-2 - 1 + 2q2 - q4 + q6

HOMFLY-PT Polynomial:

- az-1 - 5az - 8az3 - 5az5 - az7 + a3z-1 + 8a3z + 20a3z3 + 18a3z5 + 7a3z7 + a3z9 - 4a5z - 8a5z3 - 5a5z5 - a5z7

Kauffman Polynomial:

- 4a-1z3 + 4a-1z5 - a-1z7 + 3z2 - 13z4 + 12z6 - 3z8 + az-1 - 7az + 17az3 - 22az5 + 16az7 - 4az9 - a2 + 5a2z2 - 11a2z4 + 9a2z6 + 2a2z8 - 2a2z10 + a3z-1 - 11a3z + 36a3z3 - 46a3z5 + 33a3z7 - 9a3z9 + 6a4z2 - 16a4z4 + 16a4z6 - 2a4z8 - 2a4z10 - 2a5z + 2a5z3 - 5a5z5 + 9a5z7 - 5a5z9 + 2a6z2 - 12a6z4 + 14a6z6 - 7a6z8 + 2a7z - 10a7z3 + 12a7z5 - 7a7z7 - a8z2 + 5a8z4 - 5a8z6 + 3a9z3 - 3a9z5 + a10z2 - a10z4

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 = 6                       1 j = 4                     2   j = 2                   4 1   j = 0                 5 2     j = -2               8 4       j = -4             7 6         j = -6           8 7           j = -8         6 8             j = -10       4 7               j = -12     2 6                 j = -14   1 4                   j = -16   2                     j = -18 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]=	2
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 172]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 172]]
Out[4]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[16, 8, 17, 7], > X[22, 16, 7, 15], X[6, 21, 1, 22], X[18, 11, 19, 12], X[14, 6, 15, 5], > X[20, 13, 21, 14], X[12, 19, 13, 20], X[4, 18, 5, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -11, 8, -6}, > {4, -1, 2, -3, 7, -10, 9, -8, 5, -4, 11, -7, 10, -9, 6, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 3 6 10 13 15 14 13 9 q - ----- + ----- - ----- + ---- - ---- + ---- - ---- + ------- - 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q 3/2 5/2 > 6 Sqrt[q] + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 -22 2 2 3 2 5 -6 2 -2 2 4 6 -1 - q + q - --- + --- + --- - --- + -- - q + -- + q + 2 q - q + q 20 18 12 10 8 4 q q q q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 172]][a, z]
Out[8]=	3 a a 3 5 3 3 3 5 3 5 -(-) + -- - 5 a z + 8 a z - 4 a z - 8 a z + 20 a z - 8 a z - 5 a z + z z 3 5 5 5 7 3 7 5 7 3 9 > 18 a z - 5 a z - a z + 7 a z - a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 172]][a, z]
Out[9]=	3 2 a a 3 5 7 2 2 2 4 2 -a + - + -- - 7 a z - 11 a z - 2 a z + 2 a z + 3 z + 5 a z + 6 a z + z z 3 6 2 8 2 10 2 4 z 3 3 3 5 3 7 3 > 2 a z - a z + a z - ---- + 17 a z + 36 a z + 2 a z - 10 a z + a 9 3 4 2 4 4 4 6 4 8 4 10 4 > 3 a z - 13 z - 11 a z - 16 a z - 12 a z + 5 a z - a z + 5 4 z 5 3 5 5 5 7 5 9 5 6 > ---- - 22 a z - 46 a z - 5 a z + 12 a z - 3 a z + 12 z + a 7 2 6 4 6 6 6 8 6 z 7 3 7 > 9 a z + 16 a z + 14 a z - 5 a z - -- + 16 a z + 33 a z + a 5 7 7 7 8 2 8 4 8 6 8 9 3 9 > 9 a z - 7 a z - 3 z + 2 a z - 2 a z - 7 a z - 4 a z - 9 a z - 5 9 2 10 4 10 > 5 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	6 8 1 2 1 4 2 6 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 18 7 16 6 14 6 14 5 12 5 12 4 10 4 q q q t q t q t q t q t q t q t 7 6 8 8 7 7 4 t 2 2 2 > ------ + ----- + ----- + ----- + ---- + ---- + 5 t + --- + 2 t + 4 q t + 10 3 8 3 8 2 6 2 6 4 2 q t q t q t q t q t q t q 2 3 4 3 6 4 > q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a172

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