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DrawMorseLink

PD Presentation:

X10,1,11,2 X20,11,21,12 X8,9,1,10 X22,17,9,18 X12,4,13,3 X18,8,19,7 X14,6,15,5 X4,14,5,13 X6,16,7,15 X16,21,17,22 X2,20,3,19

Gauss Code:

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

Jones Polynomial:

q-9/2 - 3q-7/2 + 6q-5/2 - 9q-3/2 + 12q-1/2 - 15q1/2 + 14q3/2 - 13q5/2 + 9q7/2 - 6q9/2 + 3q11/2 - q13/2

A2 (sl(3)) Invariant:

- q-12 + q-10 - 2q-8 + q-6 - q-4 - q-2 + 3 - 2q2 + 5q4 + 3q8 + 2q10 - q12 + 2q14 - q16 + q18

HOMFLY-PT Polynomial:

- 2a-3z-1 - 5a-3z - 8a-3z3 - 5a-3z5 - a-3z7 + 3a-1z-1 + 12a-1z + 21a-1z3 + 18a-1z5 + 7a-1z7 + a-1z9 - az-1 - 5az - 8az3 - 5az5 - az7

Kauffman Polynomial:

- a-7z + 2a-7z3 - a-7z5 - 2a-6z2 + 6a-6z4 - 3a-6z6 + 6a-5z5 - 4a-5z7 + 5a-4z6 - 4a-4z8 + 2a-3z-1 - 8a-3z + 16a-3z3 - 17a-3z5 + 10a-3z7 - 4a-3z9 - 3a-2 + 11a-2z2 - 18a-2z4 + 7a-2z6 + a-2z8 - 2a-2z10 + 3a-1z-1 - 15a-1z + 39a-1z3 - 57a-1z5 + 35a-1z7 - 9a-1z9 - 3 + 17z2 - 31z4 + 17z6 - 2z10 + az-1 - 6az + 17az3 - 24az5 + 18az7 - 5az9 - a2 + 7a2z2 - 16a2z4 + 17a2z6 - 5a2z8 - 4a3z3 + 9a3z5 - 3a3z7 - a4z2 + 3a4z4 - a4z6

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 = 14                       1 j = 12                     2   j = 10                   4 1   j = 8                 6 3     j = 6               7 3       j = 4             7 6         j = 2           8 7           j = 0         5 8             j = -2       4 7               j = -4     2 5                 j = -6   1 4                   j = -8   2                     j = -10 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, 282]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 282]]
Out[4]=	PD[X[10, 1, 11, 2], X[20, 11, 21, 12], X[8, 9, 1, 10], X[22, 17, 9, 18], > X[12, 4, 13, 3], X[18, 8, 19, 7], X[14, 6, 15, 5], X[4, 14, 5, 13], > X[6, 16, 7, 15], X[16, 21, 17, 22], X[2, 20, 3, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -8, 7, -9, 6, -3}, > {3, -1, 2, -5, 8, -7, 9, -10, 4, -6, 11, -2, 10, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 3 6 9 12 3/2 5/2 q - ---- + ---- - ---- + ------- - 15 Sqrt[q] + 14 q - 13 q + 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 11/2 13/2 > 9 q - 6 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 -10 2 -6 -4 -2 2 4 8 10 12 3 - q + q - -- + q - q - q - 2 q + 5 q + 3 q + 2 q - q + 8 q 14 16 18 > 2 q - q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 282]][a, z]
Out[8]=	3 3 5 5 -2 3 a 5 z 12 z 8 z 21 z 3 5 z 18 z ---- + --- - - - --- + ---- - 5 a z - ---- + ----- - 8 a z - ---- + ----- - 3 a z z 3 a 3 a 3 a a z a a a 7 7 9 5 z 7 z 7 z > 5 a z - -- + ---- - a z + -- 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 282]][a, z]
Out[9]=	2 3 2 2 3 a z 8 z 15 z 2 2 z -3 - -- - a + ---- + --- + - - -- - --- - ---- - 6 a z + 17 z - ---- + 2 3 a z z 7 3 a 6 a a z a a a 2 3 3 3 11 z 2 2 4 2 2 z 16 z 39 z 3 3 3 > ----- + 7 a z - a z + ---- + ----- + ----- + 17 a z - 4 a z - 2 7 3 a a a a 4 4 5 5 5 5 4 6 z 18 z 2 4 4 4 z 6 z 17 z 57 z > 31 z + ---- - ----- - 16 a z + 3 a z - -- + ---- - ----- - ----- - 6 2 7 5 3 a a a a a a 6 6 6 7 5 3 5 6 3 z 5 z 7 z 2 6 4 6 4 z > 24 a z + 9 a z + 17 z - ---- + ---- + ---- + 17 a z - a z - ---- + 6 4 2 5 a a a a 7 7 8 8 9 9 10 z 35 z 7 3 7 4 z z 2 8 4 z 9 z > ----- + ----- + 18 a z - 3 a z - ---- + -- - 5 a z - ---- - ---- - 3 a 4 2 3 a a a a a 10 9 10 2 z > 5 a z - 2 z - ----- 2 a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 4 2 5 4 5 7 8 + 8 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + - + ---- + 10 5 8 4 6 4 6 3 4 3 4 2 2 2 t 2 q t q t q t q t q t q t q t q t 2 4 4 2 6 2 6 3 8 3 8 4 > 7 q t + 7 q t + 6 q t + 7 q t + 3 q t + 6 q t + 3 q t + 10 4 10 5 12 5 14 6 > 4 q t + q t + 2 q t + q t

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