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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X7,14,8,15 X15,20,16,21 X11,18,12,19 X19,12,20,13 X17,22,18,5 X21,16,22,17 X13,8,14,9 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

q-27/2 - 2q-25/2 + 4q-23/2 - 5q-21/2 + 5q-19/2 - 5q-17/2 + 4q-15/2 - 3q-13/2 - q-7/2

A2 (sl(3)) Invariant:

- q-42 - q-40 - 3q-36 + 2q-28 + 3q-24 + q-22 + 2q-20 + 3q-18 + q-16 + q-14 + q-12

HOMFLY-PT Polynomial:

- 2a7z-1 - 10a7z - 14a7z3 - 7a7z5 - a7z7 + 2a9z-1 + 6a9z + 3a9z3 + a11z-1 + 3a11z + 2a11z3 - a13z-1 - a13z

Kauffman Polynomial:

- 2a7z-1 + 10a7z - 14a7z3 + 7a7z5 - a7z7 + 3a8 - 7a8z2 + 3a8z4 - 2a9z-1 + 8a9z - 11a9z3 + 3a9z5 + 3a10z2 - 11a10z4 + 6a10z6 - a10z8 + a11z-1 - 5a11z + 9a11z3 - 10a11z5 + 5a11z7 - a11z9 - 3a12 + 14a12z2 - 20a12z4 + 13a12z6 - 3a12z8 + a13z-1 - 2a13z + a13z3 + a13z5 + 2a13z7 - a13z9 - 2a14z4 + 6a14z6 - 2a14z8 + a15z - 5a15z3 + 7a15z5 - 2a15z7 + a16 - 4a16z2 + 4a16z4 - a16z6

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 = -6                       1 j = -8                       1 j = -10                 1 1     j = -12               3         j = -14             3 3 1       j = -16           3 2           j = -18         2 3 1           j = -20       3 3               j = -22     1 2                 j = -24   1 3                   j = -26   1                     j = -28 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, 62]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 62]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[7, 14, 8, 15], X[15, 20, 16, 21], > X[11, 18, 12, 19], X[19, 12, 20, 13], X[17, 22, 18, 5], X[21, 16, 22, 17], > X[13, 8, 14, 9], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, -5, 6, -9, 3, -4, 8, -7, 5, > -6, 4, -8, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-(27/2) 2 4 5 5 5 4 3 -(7/2) q - ----- + ----- - ----- + ----- - ----- + ----- - ----- - q 25/2 23/2 21/2 19/2 17/2 15/2 13/2 q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-42 -40 3 2 3 -22 2 3 -16 -14 -12 -q - q - --- + --- + --- + q + --- + --- + q + q + q 36 28 24 20 18 q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 62]][a, z]
Out[8]=	7 9 11 13 -2 a 2 a a a 7 9 11 13 7 3 ----- + ---- + --- - --- - 10 a z + 6 a z + 3 a z - a z - 14 a z + z z z z 9 3 11 3 7 5 7 7 > 3 a z + 2 a z - 7 a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 62]][a, z]
Out[9]=	7 9 11 13 8 12 16 2 a 2 a a a 7 9 11 3 a - 3 a + a - ---- - ---- + --- + --- + 10 a z + 8 a z - 5 a z - z z z z 13 15 8 2 10 2 12 2 16 2 7 3 > 2 a z + a z - 7 a z + 3 a z + 14 a z - 4 a z - 14 a z - 9 3 11 3 13 3 15 3 8 4 10 4 12 4 > 11 a z + 9 a z + a z - 5 a z + 3 a z - 11 a z - 20 a z - 14 4 16 4 7 5 9 5 11 5 13 5 15 5 > 2 a z + 4 a z + 7 a z + 3 a z - 10 a z + a z + 7 a z + 10 6 12 6 14 6 16 6 7 7 11 7 13 7 > 6 a z + 13 a z + 6 a z - a z - a z + 5 a z + 2 a z - 15 7 10 8 12 8 14 8 11 9 13 9 > 2 a z - a z - 3 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-8 -6 1 1 1 3 1 2 3 q + q + ------- + ------- + ------- + ------ + ------ + ------ + ------ + 28 11 26 10 24 10 24 9 22 9 22 8 20 8 q t q t q t q t q t q t q t 3 2 3 3 1 2 3 3 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 20 7 18 7 18 6 16 6 18 5 16 5 14 5 14 4 q t q t q t q t q t q t q t q t 3 1 1 1 > ------ + ------ + ------ + ------ 12 4 14 3 10 3 10 2 q t q t q t q t

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