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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 3q-11/2 - 7q-9/2 + 10q-7/2 - 15q-5/2 + 16q-3/2 - 16q-1/2 + 14q1/2 - 10q3/2 + 6q5/2 - 3q7/2 + q9/2

A2 (sl(3)) Invariant:

q-20 - q-18 + q-16 + 4q-14 - q-12 + 4q-10 + 3q-8 + 3q-4 - 3q-2 + 1 - 2q2 - 2q4 + 3q6 - 2q8 + q12 - q14

HOMFLY-PT Polynomial:

a-3z + a-3z3 + a-1z-1 - a-1z3 - a-1z5 - 2az-1 - 4az - 4az3 - 2az5 - a3z - a3z3 - a3z5 + a5z-1 + a5z + a5z3

Kauffman Polynomial:

- 2a-4z2 + 3a-4z4 - a-4z6 + a-3z - 7a-3z3 + 9a-3z5 - 3a-3z7 - 2a-2 + 4a-2z2 - 7a-2z4 + 10a-2z6 - 4a-2z8 + a-1z-1 - a-1z - 2a-1z3 + 4a-1z5 + 3a-1z7 - 3a-1z9 - 5 + 25z2 - 34z4 + 25z6 - 7z8 - z10 + 2az-1 - 9az + 17az3 - 20az5 + 16az7 - 7az9 - 3a2 + 19a2z2 - 38a2z4 + 29a2z6 - 9a2z8 - a2z10 - 5a3z3 + a3z5 + 4a3z7 - 4a3z9 + a4 - 9a4z4 + 12a4z6 - 6a4z8 - a5z-1 + 7a5z - 15a5z3 + 15a5z5 - 6a5z7 + 5a6z4 - 3a6z6 + 2a7z3 - a7z5

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 = 10                       1 j = 8                     2   j = 6                   4 1   j = 4                 6 2     j = 2               8 4       j = 0             8 6         j = -2           9 9           j = -4         6 7             j = -6       4 9               j = -8     3 6                 j = -10     4                   j = -12 1 3                     j = -14 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, 187]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 187]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 3, 11, 4], X[20, 14, 21, 13], X[16, 5, 17, 6], > X[4, 15, 5, 16], X[18, 12, 19, 11], X[22, 18, 7, 17], X[14, 20, 15, 19], > X[12, 22, 13, 21], X[2, 7, 3, 8], X[6, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -5, 4, -11}, > {10, -1, 11, -2, 6, -9, 3, -8, 5, -4, 7, -6, 8, -3, 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 3 7 10 15 16 16 -q + ----- - ---- + ---- - ---- + ---- - ------- + 14 Sqrt[q] - 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 3/2 5/2 7/2 9/2 > 10 q + 6 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -18 -16 4 -12 4 3 3 3 2 4 6 1 + q - q + q + --- - q + --- + -- + -- - -- - 2 q - 2 q + 3 q - 14 10 8 4 2 q q q q q 8 12 14 > 2 q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 187]][a, z]
Out[8]=	5 3 3 1 2 a a z 3 5 z z 3 3 3 5 3 --- - --- + -- + -- - 4 a z - a z + a z + -- - -- - 4 a z - a z + a z - a z z z 3 3 a a a 5 z 5 3 5 > -- - 2 a z - a z a
In[9]:=	Kauffman[Link[11, Alternating, 187]][a, z]
Out[9]=	5 2 2 2 4 1 2 a a z z 5 2 2 z -5 - -- - 3 a + a + --- + --- - -- + -- - - - 9 a z + 7 a z + 25 z - ---- + 2 a z z z 3 a 4 a a a 2 3 3 4 z 2 2 7 z 2 z 3 3 3 5 3 7 3 > ---- + 19 a z - ---- - ---- + 17 a z - 5 a z - 15 a z + 2 a z - 2 3 a a a 4 4 5 5 4 3 z 7 z 2 4 4 4 6 4 9 z 4 z > 34 z + ---- - ---- - 38 a z - 9 a z + 5 a z + ---- + ---- - 4 2 3 a a a a 6 6 5 3 5 5 5 7 5 6 z 10 z 2 6 > 20 a z + a z + 15 a z - a z + 25 z - -- + ----- + 29 a z + 4 2 a a 7 7 4 6 6 6 3 z 3 z 7 3 7 5 7 8 > 12 a z - 3 a z - ---- + ---- + 16 a z + 4 a z - 6 a z - 7 z - 3 a a 8 9 4 z 2 8 4 8 3 z 9 3 9 10 2 10 > ---- - 9 a z - 6 a z - ---- - 7 a z - 4 a z - z - a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	9 1 1 3 4 3 6 4 9 8 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 6 12 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 6 7 9 2 2 2 4 2 4 3 > ----- + ---- + ---- + 6 t + 8 q t + 4 q t + 6 q t + 2 q t + 4 2 4 2 q t q t q t 6 3 6 4 8 4 10 5 > 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a187

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