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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-17/2 + 2q-15/2 - 4q-13/2 + 5q-11/2 - 6q-9/2 + 7q-7/2 - 7q-5/2 + 5q-3/2 - 4q-1/2 + 2q1/2 - q3/2

A2 (sl(3)) Invariant:

q-28 + 2q-26 + q-22 + q-20 - 2q-18 - q-14 - q-12 + q-10 + q-8 + 3q-6 + q-4 + 2 - q2 + q6

HOMFLY-PT Polynomial:

- a-1z - az-1 - az + az3 + a3z-1 + a3z + 2a3z3 + a5z-1 + 2a5z + 2a5z3 - 2a7z-1 - 3a7z + a9z-1

Kauffman Polynomial:

a-1z - a-1z3 + z2 - 2z4 + az-1 - 3az + 3az3 - 3az5 - a2 + a2z2 + 2a2z4 - 3a2z6 + a3z-1 - 4a3z + 4a3z3 + 3a3z5 - 3a3z7 - 2a4 + 7a4z2 - 8a4z4 + 8a4z6 - 3a4z8 - a5z-1 + 8a5z - 18a5z3 + 16a5z5 - 2a5z7 - a5z9 - 3a6 + 11a6z2 - 23a6z4 + 20a6z6 - 5a6z8 - 2a7z-1 + 13a7z - 26a7z3 + 15a7z5 - a7z9 - a8 + 4a8z2 - 11a8z4 + 9a8z6 - 2a8z8 - a9z-1 + 5a9z - 8a9z3 + 5a9z5 - a9z7

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

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