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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-12 - 3q-11 + 6q-10 - 9q-9 + 12q-8 - 12q-7 + 13q-6 - 9q-5 + 7q-4 - 3q-3 + q-2

A2 (sl(3)) Invariant:

q-38 + q-36 - 2q-34 + q-32 + q-30 - q-28 + 5q-26 + 3q-24 + 5q-22 + 5q-20 + 2q-18 + 5q-16 - q-14 + q-12 + 2q-10 - 2q-8 + q-6

HOMFLY-PT Polynomial:

a4z2 + a4z4 + a6z-2 + 4a6 + 6a6z2 + 3a6z4 - 2a8z-2 - 3a8 + a8z2 + 2a8z4 + a10z-2 - 2a10 - 3a10z2 + a12

Kauffman Polynomial:

- a4z2 + a4z4 - 2a5z3 + 3a5z5 + a6z-2 - 4a6 + 9a6z2 - 9a6z4 + 6a6z6 - 2a7z-1 + 4a7z - a7z3 - 5a7z5 + 6a7z7 + 2a8z-2 - 6a8 + 12a8z2 - 13a8z4 + 2a8z6 + 4a8z8 - 2a9z-1 + 15a9z3 - 26a9z5 + 11a9z7 + a9z9 + a10z-2 - 3a10 + 7a10z2 - 6a10z4 - 9a10z6 + 7a10z8 - 6a11z + 22a11z3 - 27a11z5 + 8a11z7 + a11z9 - a12 + 8a12z2 - 6a12z4 - 4a12z6 + 3a12z8 - 2a13z + 8a13z3 - 9a13z5 + 3a13z7 - a14 + 3a14z2 - 3a14z4 + a14z6

Khovanov Homology:

trqj r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -3                     1 j = -5                   3 1 j = -7                 4     j = -9               5 3     j = -11             8 4       j = -13           5 6         j = -15         7 7           j = -17       4 7             j = -19     2 5               j = -21   1 4                 j = -23   2                   j = -25 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]=	3
In[3]:=	Show[DrawMorseLink[Link[10, Alternating, 130]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 130]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[10, 13, 5, 14], X[20, 15, 11, 16], > X[14, 7, 15, 8], X[8, 17, 9, 18], X[18, 9, 19, 10], X[16, 19, 17, 20], > X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 5, -6, 7, -3}, > {10, -2, 3, -5, 4, -8, 6, -7, 8, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-12 3 6 9 12 12 13 9 7 3 -2 q - --- + --- - -- + -- - -- + -- - -- + -- - -- + q 11 10 9 8 7 6 5 4 3 q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-38 -36 2 -32 -30 -28 5 3 5 5 2 5 q + q - --- + q + q - q + --- + --- + --- + --- + --- + --- - 34 26 24 22 20 18 16 q q q q q q q -14 -12 2 2 -6 > q + q + --- - -- + q 10 8 q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 130]][a, z]
Out[8]=	6 8 10 6 8 10 12 a 2 a a 4 2 6 2 8 2 4 a - 3 a - 2 a + a + -- - ---- + --- + a z + 6 a z + a z - 2 2 2 z z z 10 2 4 4 6 4 8 4 > 3 a z + a z + 3 a z + 2 a z
In[9]:=	Kauffman[Link[10, Alternating, 130]][a, z]
Out[9]=	6 8 10 7 9 6 8 10 12 14 a 2 a a 2 a 2 a 7 -4 a - 6 a - 3 a - a - a + -- + ---- + --- - ---- - ---- + 4 a z - 2 2 2 z z z z z 11 13 4 2 6 2 8 2 10 2 12 2 > 6 a z - 2 a z - a z + 9 a z + 12 a z + 7 a z + 8 a z + 14 2 5 3 7 3 9 3 11 3 13 3 4 4 > 3 a z - 2 a z - a z + 15 a z + 22 a z + 8 a z + a z - 6 4 8 4 10 4 12 4 14 4 5 5 7 5 > 9 a z - 13 a z - 6 a z - 6 a z - 3 a z + 3 a z - 5 a z - 9 5 11 5 13 5 6 6 8 6 10 6 12 6 > 26 a z - 27 a z - 9 a z + 6 a z + 2 a z - 9 a z - 4 a z + 14 6 7 7 9 7 11 7 13 7 8 8 10 8 > a z + 6 a z + 11 a z + 8 a z + 3 a z + 4 a z + 7 a z + 12 8 9 9 11 9 > 3 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	-5 -3 1 2 1 4 2 5 4 q + q + ------- + ------ + ------ + ------ + ------ + ------ + ------ + 25 10 23 9 21 9 21 8 19 8 19 7 17 7 q t q t q t q t q t q t q t 7 7 7 5 6 8 4 5 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 17 6 15 6 15 5 13 5 13 4 11 4 11 3 9 3 q t q t q t q t q t q t q t q t 3 4 3 > ----- + ----- + ---- 9 2 7 2 5 q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a130

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