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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-21/2 - 3q-19/2 + 7q-17/2 - 14q-15/2 + 17q-13/2 - 23q-11/2 + 21q-9/2 - 21q-7/2 + 14q-5/2 - 10q-3/2 + 4q-1/2 - q1/2

A2 (sl(3)) Invariant:

- q-34 - 2q-32 + q-30 + q-28 + 10q-24 + 7q-22 + 9q-20 + 15q-18 + 8q-16 + 13q-14 + 5q-12 + 6q-10 + 7q-8 - 2q-6 + 5q-4 - 2 + q2

HOMFLY-PT Polynomial:

- az3 - a3z-3 - 4a3z-1 - 6a3z - 2a3z3 + a3z5 + 3a5z-3 + 10a5z-1 + 13a5z + 8a5z3 + 3a5z5 - 3a7z-3 - 9a7z-1 - 11a7z - 6a7z3 + a9z-3 + 4a9z-1 + 4a9z - a11z-1

Kauffman Polynomial:

az3 - az5 + 4a2z4 - 4a2z6 + a3z-3 - 5a3z-1 + 10a3z - 14a3z3 + 16a3z5 - 9a3z7 - 3a4z-2 + 9a4 - 7a4z2 - 4a4z4 + 14a4z6 - 10a4z8 + 3a5z-3 - 14a5z-1 + 35a5z - 56a5z3 + 48a5z5 - 12a5z7 - 5a5z9 - 6a6z-2 + 21a6 - 24a6z2 - 9a6z4 + 37a6z6 - 19a6z8 - a6z10 + 3a7z-3 - 18a7z-1 + 49a7z - 74a7z3 + 56a7z5 - 7a7z7 - 8a7z9 - 3a8z-2 + 18a8 - 28a8z2 + 4a8z4 + 24a8z6 - 13a8z8 - a8z10 + a9z-3 - 11a9z-1 + 30a9z - 42a9z3 + 33a9z5 - 7a9z7 - 3a9z9 + 6a10 - 14a10z2 + 8a10z4 + 4a10z6 - 4a10z8 - 2a11z-1 + 6a11z - 9a11z3 + 8a11z5 - 3a11z7 + a12 - 3a12z2 + 3a12z4 - a12z6

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 2                       1 j = 0                     3   j = -2                   7 1   j = -4                 8 4     j = -6               13 6       j = -8             11 11         j = -10           12 10           j = -12         8 14             j = -14       6 9               j = -16     2 9                 j = -18   1 5                   j = -20   2                     j = -22 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]=	4
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 534]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 534]]
Out[4]=	PD[X[6, 1, 7, 2], X[2, 5, 3, 6], X[18, 11, 19, 12], X[10, 3, 11, 4], > X[4, 9, 1, 10], X[16, 7, 17, 8], X[8, 15, 5, 16], X[20, 14, 21, 13], > X[22, 19, 15, 20], X[12, 22, 13, 21], X[14, 17, 9, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, 3, -10, 8, -11}, > {7, -6, 11, -3, 9, -8, 10, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 3 7 14 17 23 21 21 14 10 q - ----- + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q q 4 > ------- - Sqrt[q] Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-34 2 -30 -28 10 7 9 15 8 13 5 6 -2 - q - --- + q + q + --- + --- + --- + --- + --- + --- + --- + --- + 32 24 22 20 18 16 14 12 10 q q q q q q q q q 7 2 5 2 > -- - -- + -- + q 8 6 4 q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 534]][a, z]
Out[8]=	3 5 7 9 3 5 7 9 11 a 3 a 3 a a 4 a 10 a 9 a 4 a a 3 -(--) + ---- - ---- + -- - ---- + ----- - ---- + ---- - --- - 6 a z + 3 3 3 3 z z z z z z z z z 5 7 9 3 3 3 5 3 7 3 3 5 > 13 a z - 11 a z + 4 a z - a z - 2 a z + 8 a z - 6 a z + a z + 5 5 > 3 a z
In[9]:=	Kauffman[Link[11, Alternating, 534]][a, z]
Out[9]=	3 5 7 9 4 6 4 6 8 10 12 a 3 a 3 a a 3 a 6 a 9 a + 21 a + 18 a + 6 a + a + -- + ---- + ---- + -- - ---- - ---- - 3 3 3 3 2 2 z z z z z z 8 3 5 7 9 11 3 a 5 a 14 a 18 a 11 a 2 a 3 5 7 > ---- - ---- - ----- - ----- - ----- - ----- + 10 a z + 35 a z + 49 a z + 2 z z z z z z 9 11 4 2 6 2 8 2 10 2 12 2 > 30 a z + 6 a z - 7 a z - 24 a z - 28 a z - 14 a z - 3 a z + 3 3 3 5 3 7 3 9 3 11 3 2 4 > a z - 14 a z - 56 a z - 74 a z - 42 a z - 9 a z + 4 a z - 4 4 6 4 8 4 10 4 12 4 5 3 5 > 4 a z - 9 a z + 4 a z + 8 a z + 3 a z - a z + 16 a z + 5 5 7 5 9 5 11 5 2 6 4 6 6 6 > 48 a z + 56 a z + 33 a z + 8 a z - 4 a z + 14 a z + 37 a z + 8 6 10 6 12 6 3 7 5 7 7 7 9 7 > 24 a z + 4 a z - a z - 9 a z - 12 a z - 7 a z - 7 a z - 11 7 4 8 6 8 8 8 10 8 5 9 7 9 > 3 a z - 10 a z - 19 a z - 13 a z - 4 a z - 5 a z - 8 a z - 9 9 6 10 8 10 > 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 7 1 2 1 5 2 9 6 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 22 9 20 8 18 8 18 7 16 7 16 6 14 6 q q q t q t q t q t q t q t q t 9 8 14 12 10 11 11 13 6 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 14 5 12 5 12 4 10 4 10 3 8 3 8 2 6 2 6 q t q t q t q t q t q t q t q t q t 8 t 2 2 > ---- + 3 t + -- + q t 4 2 q t q

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