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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-19/2 + 3q-17/2 - 8q-15/2 + 11q-13/2 - 16q-11/2 + 15q-9/2 - 17q-7/2 + 11q-5/2 - 9q-3/2 + 4q-1/2 - q1/2

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

- az3 - a3z-3 - 4a3z-1 - 5a3z - a3z3 + a3z5 + 3a5z-3 + 9a5z-1 + 9a5z + 5a5z3 + 2a5z5 - 3a7z-3 - 6a7z-1 - 5a7z - 3a7z3 + a9z-3 + a9z-1 + a9z

Kauffman Polynomial:

az3 - az5 + 5a2z4 - 4a2z6 + a3z-3 - 5a3z-1 + 7a3z - 12a3z3 + 16a3z5 - 8a3z7 - 3a4z-2 + 10a4 - 7a4z2 + 9a4z6 - 7a4z8 + 3a5z-3 - 12a5z-1 + 23a5z - 39a5z3 + 40a5z5 - 14a5z7 - 2a5z9 - 6a6z-2 + 19a6 - 23a6z2 + 4a6z4 + 17a6z6 - 12a6z8 + 3a7z-3 - 12a7z-1 + 29a7z - 42a7z3 + 36a7z5 - 12a7z7 - 2a7z9 - 3a8z-2 + 10a8 - 17a8z2 + 13a8z4 + a8z6 - 5a8z8 + a9z-3 - 5a9z-1 + 12a9z - 14a9z3 + 12a9z5 - 6a9z7 - a10z2 + 4a10z4 - 3a10z6 - a11z + 2a11z3 - a11z5

Khovanov Homology:

trqj 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                 6 1   j = -4               6 4     j = -6             11 5       j = -8           8 10         j = -10         8 7           j = -12       5 10             j = -14     3 6               j = -16     5                 j = -18 1 3                   j = -20 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[10, Alternating, 173]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 173]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[20, 13, 17, 14], X[16, 19, 11, 20], > X[18, 7, 19, 8], X[8, 16, 9, 15], X[14, 10, 15, 9], X[10, 17, 5, 18], > X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {8, -5, 4, -3}, {9, -1, 5, -6, 7, -8}, > {10, -2, 3, -7, 6, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 3 8 11 16 15 17 11 9 -q + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q 4 > ------- - Sqrt[q] Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 -28 7 6 7 14 9 13 8 6 7 -6 -2 + q + q + --- + --- + --- + --- + --- + --- + --- + --- + -- - q + 24 22 20 18 16 14 12 10 8 q q q q q q q q q 4 2 > -- + q 4 q
In[8]:=	HOMFLYPT[Link[10, Alternating, 173]][a, z]
Out[8]=	3 5 7 9 3 5 7 9 a 3 a 3 a a 4 a 9 a 6 a a 3 5 7 -(--) + ---- - ---- + -- - ---- + ---- - ---- + -- - 5 a z + 9 a z - 5 a z + 3 3 3 3 z z z z z z z z 9 3 3 3 5 3 7 3 3 5 5 5 > a z - a z - a z + 5 a z - 3 a z + a z + 2 a z
In[9]:=	Kauffman[Link[10, Alternating, 173]][a, z]
Out[9]=	3 5 7 9 4 6 8 3 4 6 8 a 3 a 3 a a 3 a 6 a 3 a 5 a 10 a + 19 a + 10 a + -- + ---- + ---- + -- - ---- - ---- - ---- - ---- - 3 3 3 3 2 2 2 z z z z z z z z 5 7 9 12 a 12 a 5 a 3 5 7 9 11 > ----- - ----- - ---- + 7 a z + 23 a z + 29 a z + 12 a z - a z - z z z 4 2 6 2 8 2 10 2 3 3 3 5 3 > 7 a z - 23 a z - 17 a z - a z + a z - 12 a z - 39 a z - 7 3 9 3 11 3 2 4 6 4 8 4 10 4 > 42 a z - 14 a z + 2 a z + 5 a z + 4 a z + 13 a z + 4 a z - 5 3 5 5 5 7 5 9 5 11 5 2 6 > a z + 16 a z + 40 a z + 36 a z + 12 a z - a z - 4 a z + 4 6 6 6 8 6 10 6 3 7 5 7 7 7 > 9 a z + 17 a z + a z - 3 a z - 8 a z - 14 a z - 12 a z - 9 7 4 8 6 8 8 8 5 9 7 9 > 6 a z - 7 a z - 12 a z - 5 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 6 1 1 3 5 3 6 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 20 8 18 8 18 7 16 6 14 6 14 5 12 5 q q q t q t q t q t q t q t q t 10 8 7 8 10 11 5 6 t > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + 3 t + -- + 12 4 10 4 10 3 8 3 8 2 6 2 6 4 2 q t q t q t q t q t q t q t q t q 2 2 > q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a173

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