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DrawMorseLink

PD Presentation:

X6172 X3,10,4,11 X7,20,8,21 X15,5,16,22 X21,17,22,16 X9,14,10,15 X13,19,14,18 X19,13,20,12 X17,8,18,9 X2536 X11,4,12,1

Gauss Code:

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

Jones Polynomial:

q-13/2 - 5q-11/2 + 7q-9/2 - 9q-7/2 + 10q-5/2 - 10q-3/2 + 8q-1/2 - 6q1/2 + 3q3/2 - q5/2

A2 (sl(3)) Invariant:

- q-24 + q-22 + 3q-18 + 2q-16 + 3q-12 - 2q-10 + 2q-8 - q-6 - q-4 + q-2 - 1 + 2q2 + q8

HOMFLY-PT Polynomial:

- a-1z-1 - 2a-1z - a-1z3 + 3az-1 + 7az + 7az3 + 2az5 - 4a3z-1 - 11a3z - 10a3z3 - 5a3z5 - a3z7 + 2a5z-1 + 5a5z + 4a5z3 + a5z5 - a7z

Kauffman Polynomial:

- a-1z-1 + 4a-1z - 6a-1z3 + 4a-1z5 - a-1z7 - 1 + 5z2 - 14z4 + 12z6 - 3z8 - 3az-1 + 17az - 33az3 + 20az5 + az7 - 2az9 - 3a2 + 16a2z2 - 38a2z4 + 37a2z6 - 10a2z8 - 4a3z-1 + 26a3z - 53a3z3 + 42a3z5 - 6a3z7 - 2a3z9 - 2a4 + 12a4z2 - 22a4z4 + 22a4z6 - 7a4z8 - 2a5z-1 + 14a5z - 31a5z3 + 26a5z5 - 8a5z7 - a6 + 2a6z4 - 3a6z6 + a7z - 5a7z3 - a8z2

Khovanov Homology:

trqj 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             4 2     j = -2           6 4       j = -4         5 5         j = -6       4 5           j = -8     3 5             j = -10   2 4               j = -12   4                 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, NonAlternating, 77]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 77]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 10, 4, 11], X[7, 20, 8, 21], X[15, 5, 16, 22], > X[21, 17, 22, 16], X[9, 14, 10, 15], X[13, 19, 14, 18], X[19, 13, 20, 12], > X[17, 8, 18, 9], X[2, 5, 3, 6], X[11, 4, 12, 1]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, -2, 11}, {10, -1, -3, 9, -6, 2, -11, 8, -7, 6, -4, 5, -9, 7, > -8, 3, -5, 4}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 5 7 9 10 10 8 3/2 q - ----- + ---- - ---- + ---- - ---- + ------- - 6 Sqrt[q] + 3 q - 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 5/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 -22 3 2 3 2 2 -6 -4 -2 2 8 -1 - q + q + --- + --- + --- - --- + -- - q - q + q + 2 q + q 18 16 12 10 8 q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 77]][a, z]
Out[8]=	3 5 3 1 3 a 4 a 2 a 2 z 3 5 7 z -(---) + --- - ---- + ---- - --- + 7 a z - 11 a z + 5 a z - a z - -- + a z z z z a a 3 3 3 5 3 5 3 5 5 5 3 7 > 7 a z - 10 a z + 4 a z + 2 a z - 5 a z + a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 77]][a, z]
Out[9]=	3 5 2 4 6 1 3 a 4 a 2 a 4 z 3 -1 - 3 a - 2 a - a - --- - --- - ---- - ---- + --- + 17 a z + 26 a z + a z z z z a 3 5 7 2 2 2 4 2 8 2 6 z 3 > 14 a z + a z + 5 z + 16 a z + 12 a z - a z - ---- - 33 a z - a 3 3 5 3 7 3 4 2 4 4 4 6 4 > 53 a z - 31 a z - 5 a z - 14 z - 38 a z - 22 a z + 2 a z + 5 4 z 5 3 5 5 5 6 2 6 4 6 > ---- + 20 a z + 42 a z + 26 a z + 12 z + 37 a z + 22 a z - a 7 6 6 z 7 3 7 5 7 8 2 8 4 8 > 3 a z - -- + a z - 6 a z - 8 a z - 3 z - 10 a z - 7 a z - a 9 3 9 > 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 6 1 4 2 4 3 5 4 5 -- + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 4 2 14 5 12 4 10 4 10 3 8 3 8 2 6 2 6 q q q t q t q t q t q t q t q t q t 5 4 t 2 2 2 2 3 4 3 6 4 > ---- + 4 t + --- + 2 t + 4 q t + q t + 2 q t + q t 4 2 q t q

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