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DrawMorseLink

PD Presentation:

X6172 X10,4,11,3 X18,8,19,7 X20,13,21,14 X22,15,5,16 X14,21,15,22 X16,19,17,20 X12,10,13,9 X8,18,9,17 X2536 X4,12,1,11

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 3q-11/2 - 7q-9/2 + 12q-7/2 - 16q-5/2 + 18q-3/2 - 18q-1/2 + 15q1/2 - 12q3/2 + 6q5/2 - 3q7/2 + q9/2

A2 (sl(3)) Invariant:

q-20 + 3q-14 - 3q-12 + q-10 - 3q-6 + 3q-4 - 3q-2 + 4 + q2 + q4 + 5q6 - q8 + q10 - q14

HOMFLY-PT Polynomial:

2a-3z + a-3z3 - 2a-1z-1 - 6a-1z - 6a-1z3 - 2a-1z5 + 4az-1 + 9az + 8az3 + 4az5 + az7 - 3a3z-1 - 7a3z - 6a3z3 - 2a3z5 + a5z-1 + 2a5z + a5z3

Kauffman Polynomial:

- 2a-4z2 + 3a-4z4 - a-4z6 + 4a-3z - 9a-3z3 + 9a-3z5 - 3a-3z7 - 2a-2z2 - 2a-2z4 + 8a-2z6 - 4a-2z8 - 2a-1z-1 + 13a-1z - 24a-1z3 + 18a-1z5 - a-1z7 - 3a-1z9 + 2 - 3z2 - 7z4 + 18z6 - 8z8 - z10 - 4az-1 + 19az - 33az3 + 24az5 - 6az9 + 3a2 - 8a2z2 + 14a2z6 - 9a2z8 - a2z10 - 3a3z-1 + 13a3z - 24a3z3 + 23a3z5 - 7a3z7 - 3a3z9 + 3a4 - 8a4z2 + 7a4z4 + 2a4z6 - 5a4z8 - a5z-1 + 2a5z - 4a5z3 + 7a5z5 - 5a5z7 + a6 - 3a6z2 + 5a6z4 - 3a6z6 - a7z + 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                 8 2     j = 2               7 4       j = 0             11 8         j = -2           9 9           j = -4         7 9             j = -6       5 9               j = -8     2 7                 j = -10   1 5                   j = -12   2                     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, 42]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 42]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[18, 8, 19, 7], X[20, 13, 21, 14], > X[22, 15, 5, 16], X[14, 21, 15, 22], X[16, 19, 17, 20], X[12, 10, 13, 9], > X[8, 18, 9, 17], X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -9, 8, -2, 11, -8, 4, -6, 5, -7, 9, -3, > 7, -4, 6, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 3 7 12 16 18 18 -q + ----- - ---- + ---- - ---- + ---- - ------- + 15 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 > 12 q + 6 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 3 3 -10 3 3 3 2 4 6 8 10 14 4 + q + --- - --- + q - -- + -- - -- + q + q + 5 q - q + q - q 14 12 6 4 2 q q q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 42]][a, z]
Out[8]=	3 5 3 3 -2 4 a 3 a a 2 z 6 z 3 5 z 6 z --- + --- - ---- + -- + --- - --- + 9 a z - 7 a z + 2 a z + -- - ---- + a z z z z 3 a 3 a a a 5 3 3 3 5 3 2 z 5 3 5 7 > 8 a z - 6 a z + a z - ---- + 4 a z - 2 a z + a z a
In[9]:=	Kauffman[Link[11, Alternating, 42]][a, z]
Out[9]=	3 5 2 4 6 2 4 a 3 a a 4 z 13 z 3 2 + 3 a + 3 a + a - --- - --- - ---- - -- + --- + ---- + 19 a z + 13 a z + a z z z z 3 a a 2 2 3 5 7 2 2 z 2 z 2 2 4 2 6 2 9 z > 2 a z - a z - 3 z - ---- - ---- - 8 a z - 8 a z - 3 a z - ---- - 4 2 3 a a a 3 4 4 24 z 3 3 3 5 3 7 3 4 3 z 2 z > ----- - 33 a z - 24 a z - 4 a z + 2 a z - 7 z + ---- - ---- + a 4 2 a a 5 5 4 4 6 4 9 z 18 z 5 3 5 5 5 7 5 > 7 a z + 5 a z + ---- + ----- + 24 a z + 23 a z + 7 a z - a z + 3 a a 6 6 7 7 6 z 8 z 2 6 4 6 6 6 3 z z 3 7 > 18 z - -- + ---- + 14 a z + 2 a z - 3 a z - ---- - -- - 7 a z - 4 2 3 a a a a 8 9 5 7 8 4 z 2 8 4 8 3 z 9 3 9 10 > 5 a z - 8 z - ---- - 9 a z - 5 a z - ---- - 6 a z - 3 a z - z - 2 a a 2 10 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	9 1 2 1 5 2 7 5 9 11 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 5 10 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 7 9 9 2 2 2 4 2 4 3 > ----- + ---- + ---- + 8 t + 7 q t + 4 q t + 8 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

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