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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-25/2 - 2q-23/2 + 4q-21/2 - 5q-19/2 + 6q-17/2 - 8q-15/2 + 7q-13/2 - 7q-11/2 + 5q-9/2 - 4q-7/2 + 2q-5/2 - q-3/2

A2 (sl(3)) Invariant:

- q-40 - 2q-38 - q-34 - q-32 + 2q-30 + q-26 + 2q-24 + q-22 + 2q-20 + q-18 + 2q-16 + 2q-14 - q-12 + q-10 + q-8 - q-6 + q-4

HOMFLY-PT Polynomial:

- a3z - a3z3 - a5z-1 - 2a5z - 2a5z3 + a7z-1 - 2a7z3 - a9z-1 - 2a9z - 2a9z3 + 2a11z-1 + 3a11z - a13z-1

Kauffman Polynomial:

a3z - a3z3 + a4z2 - 2a4z4 + a5z-1 - 3a5z + 3a5z3 - 3a5z5 - a6 + a6z2 + 2a6z4 - 3a6z6 + a7z-1 - 4a7z + 4a7z3 + 3a7z5 - 3a7z7 + 6a8z6 - 3a8z8 + a9z-1 - 5a9z + 13a9z3 - 14a9z5 + 11a9z7 - 3a9z9 - 4a10 + 18a10z2 - 31a10z4 + 19a10z6 - a10z8 - a10z10 + 2a11z-1 - 8a11z + 23a11z3 - 38a11z5 + 25a11z7 - 5a11z9 - 7a12 + 28a12z2 - 39a12z4 + 16a12z6 + a12z8 - a12z10 + a13z-1 - 3a13z + 10a13z3 - 18a13z5 + 11a13z7 - 2a13z9 - 3a14 + 10a14z2 - 12a14z4 + 6a14z6 - a14z8

Khovanov Homology:

trqj r = -11 r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -2                       1 j = -4                     2 1 j = -6                   2     j = -8                 3 2     j = -10               4 2       j = -12             4 4         j = -14           4 3           j = -16         2 4             j = -18       3 4               j = -20     1 2                 j = -22   1 3                   j = -24   1                     j = -26 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, 37]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 37]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[18, 15, 19, 16], X[16, 7, 17, 8], > X[8, 17, 9, 18], X[22, 11, 5, 12], X[20, 13, 21, 14], X[14, 19, 15, 20], > X[12, 21, 13, 22], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 4, -5, 11, -2, 6, -9, 7, -8, 3, -4, 5, -3, > 8, -7, 9, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(25/2) 2 4 5 6 8 7 7 5 q - ----- + ----- - ----- + ----- - ----- + ----- - ----- + ---- - 23/2 21/2 19/2 17/2 15/2 13/2 11/2 9/2 q q q q q q q q 4 2 -(3/2) > ---- + ---- - q 7/2 5/2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-40 2 -34 -32 2 -26 2 -22 2 -18 2 2 -q - --- - q - q + --- + q + --- + q + --- + q + --- + --- - 38 30 24 20 16 14 q q q q q q -12 -10 -8 -6 -4 > q + q + q - q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 37]][a, z]
Out[8]=	5 7 9 11 13 a a a 2 a a 3 5 9 11 3 3 -(--) + -- - -- + ----- - --- - a z - 2 a z - 2 a z + 3 a z - a z - z z z z z 5 3 7 3 9 3 > 2 a z - 2 a z - 2 a z
In[9]:=	Kauffman[Link[11, Alternating, 37]][a, z]
Out[9]=	5 7 9 11 13 6 10 12 14 a a a 2 a a 3 5 -a - 4 a - 7 a - 3 a + -- + -- + -- + ----- + --- + a z - 3 a z - z z z z z 7 9 11 13 4 2 6 2 10 2 > 4 a z - 5 a z - 8 a z - 3 a z + a z + a z + 18 a z + 12 2 14 2 3 3 5 3 7 3 9 3 11 3 > 28 a z + 10 a z - a z + 3 a z + 4 a z + 13 a z + 23 a z + 13 3 4 4 6 4 10 4 12 4 14 4 > 10 a z - 2 a z + 2 a z - 31 a z - 39 a z - 12 a z - 5 5 7 5 9 5 11 5 13 5 6 6 8 6 > 3 a z + 3 a z - 14 a z - 38 a z - 18 a z - 3 a z + 6 a z + 10 6 12 6 14 6 7 7 9 7 11 7 > 19 a z + 16 a z + 6 a z - 3 a z + 11 a z + 25 a z + 13 7 8 8 10 8 12 8 14 8 9 9 11 9 > 11 a z - 3 a z - a z + a z - a z - 3 a z - 5 a z - 13 9 10 10 12 10 > 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-4 -2 1 1 1 3 1 2 3 q + q + ------- + ------- + ------- + ------ + ------ + ------ + ------ + 26 11 24 10 22 10 22 9 20 9 20 8 18 8 q t q t q t q t q t q t q t 4 2 4 4 3 4 4 4 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 18 7 16 7 16 6 14 6 14 5 12 5 12 4 10 4 q t q t q t q t q t q t q t q t 2 3 2 2 2 > ------ + ----- + ----- + ----- + ---- 10 3 8 3 8 2 6 2 4 q t q t q t q t q t

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