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Knotscape

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DrawMorseLink

PD Presentation:

X8192 X18,9,19,10 X14,6,15,5 X16,12,17,11 X10,4,11,3 X12,7,13,8 X20,15,7,16 X6,14,1,13 X4,19,5,20 X2,18,3,17

Gauss Code:

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

Jones Polynomial:

q-11/2 - 4q-9/2 + 9q-7/2 - 14q-5/2 + 17q-3/2 - 19q-1/2 + 17q1/2 - 14q3/2 + 9q5/2 - 5q7/2 + q9/2

A2 (sl(3)) Invariant:

- q-16 + 2q-14 - 3q-12 + q-10 + 3q-8 - 3q-6 + 4q-4 - 3q-2 + 2 + q2 - q4 + 5q6 - q8 + 2q10 + 2q12 - q14

HOMFLY-PT Polynomial:

- a-3z-1 + a-3z3 + a-1z-1 - a-1z - 4a-1z3 - 2a-1z5 + 4az + 7az3 + 4az5 + az7 - 2a3z - 2a3z3 - a3z5

Kauffman Polynomial:

a-4z4 - a-4z6 + a-3z-1 - 2a-3z - 5a-3z3 + 11a-3z5 - 5a-3z7 - a-2 + a-2z2 - 11a-2z4 + 19a-2z6 - 8a-2z8 + a-1z-1 - 19a-1z3 + 30a-1z5 - 5a-1z7 - 4a-1z9 + 3z2 - 22z4 + 42z6 - 19z8 + 5az - 27az3 + 42az5 - 13az7 - 4az9 - a2z2 - a2z4 + 13a2z6 - 11a2z8 + 3a3z - 12a3z3 + 19a3z5 - 13a3z7 - 3a4z2 + 8a4z4 - 9a4z6 + a5z3 - 4a5z5 - a6z4

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 10                     1 j = 8                   4   j = 6                 5 1   j = 4               9 4     j = 2             9 6       j = 0           10 8         j = -2         8 10           j = -4       6 9             j = -6     3 8               j = -8   1 6                 j = -10   3                   j = -12 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, 70]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 70]]
Out[4]=	PD[X[8, 1, 9, 2], X[18, 9, 19, 10], X[14, 6, 15, 5], X[16, 12, 17, 11], > X[10, 4, 11, 3], X[12, 7, 13, 8], X[20, 15, 7, 16], X[6, 14, 1, 13], > X[4, 19, 5, 20], X[2, 18, 3, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 5, -9, 3, -8}, > {6, -1, 2, -5, 4, -6, 8, -3, 7, -4, 10, -2, 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 4 9 14 17 19 3/2 q - ---- + ---- - ---- + ---- - ------- + 17 Sqrt[q] - 14 q + 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 9/2 > 9 q - 5 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 2 3 -10 3 3 4 3 2 4 6 8 10 2 - q + --- - --- + q + -- - -- + -- - -- + q - q + 5 q - q + 2 q + 14 12 8 6 4 2 q q q q q q 12 14 > 2 q - q
In[8]:=	HOMFLYPT[Link[10, Alternating, 70]][a, z]
Out[8]=	3 3 5 1 1 z 3 z 4 z 3 3 3 2 z -(----) + --- - - + 4 a z - 2 a z + -- - ---- + 7 a z - 2 a z - ---- + 3 a z a 3 a a a z a 5 3 5 7 > 4 a z - a z + a z
In[9]:=	Kauffman[Link[10, Alternating, 70]][a, z]
Out[9]=	2 3 -2 1 1 2 z 3 2 z 2 2 4 2 5 z -a + ---- + --- - --- + 5 a z + 3 a z + 3 z + -- - a z - 3 a z - ---- - 3 a z 3 2 3 a z a a a 3 4 4 19 z 3 3 3 5 3 4 z 11 z 2 4 4 4 > ----- - 27 a z - 12 a z + a z - 22 z + -- - ----- - a z + 8 a z - a 4 2 a a 5 5 6 6 6 4 11 z 30 z 5 3 5 5 5 6 z 19 z > a z + ----- + ----- + 42 a z + 19 a z - 4 a z + 42 z - -- + ----- + 3 a 4 2 a a a 7 7 8 2 6 4 6 5 z 5 z 7 3 7 8 8 z > 13 a z - 9 a z - ---- - ---- - 13 a z - 13 a z - 19 z - ---- - 3 a 2 a a 9 2 8 4 z 9 > 11 a z - ---- - 4 a z a
In[10]:=	Kh[L][q, t]
Out[10]=	10 1 3 1 6 3 8 6 9 10 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 2 12 5 10 4 8 4 8 3 6 3 6 2 4 2 4 q q t q t q t q t q t q t q t q t 8 2 2 2 4 2 4 3 6 3 6 4 > ---- + 8 t + 9 q t + 6 q t + 9 q t + 4 q t + 5 q t + q t + 2 q t 8 4 10 5 > 4 q t + q t

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