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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 5q-11/2 - 11q-9/2 + 16q-7/2 - 21q-5/2 + 22q-3/2 - 21q-1/2 + 16q1/2 - 11q3/2 + 5q5/2 - q7/2

A2 (sl(3)) Invariant:

q-20 - 2q-18 - q-16 + 4q-14 - 3q-12 + 4q-10 + q-8 - q-6 + 5q-4 - 3q-2 + 6 - 2q2 - q4 + 3q6 - 3q8 + q10

HOMFLY-PT Polynomial:

- a-1z3 - a-1z5 - az-1 - az + 3az3 + 3az5 + az7 + a3z-1 - 3a3z3 - 2a3z5 + a5z3

Kauffman Polynomial:

- a-3z5 + 4a-2z4 - 5a-2z6 - 6a-1z3 + 16a-1z5 - 11a-1z7 - 4z4 + 16z6 - 12z8 + az-1 - az - 18az3 + 39az5 - 14az7 - 5az9 - a2 - 16a2z4 + 42a2z6 - 24a2z8 + a3z-1 - a3z - 18a3z3 + 39a3z5 - 14a3z7 - 5a3z9 - 4a4z4 + 16a4z6 - 12a4z8 - 6a5z3 + 16a5z5 - 11a5z7 + 4a6z4 - 5a6z6 - a7z5

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 8                     1 j = 6                   4   j = 4                 7 1   j = 2               9 4     j = 0             12 7       j = -2           11 10         j = -4         10 11           j = -6       7 12             j = -8     4 9               j = -10   1 7                 j = -12   4                   j = -14 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, 56]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 56]]
Out[4]=	PD[X[8, 1, 9, 2], X[16, 7, 17, 8], X[10, 4, 11, 3], X[12, 5, 13, 6], > X[20, 12, 7, 11], X[18, 13, 19, 14], X[2, 15, 3, 16], X[4, 19, 5, 20], > X[14, 10, 15, 9], X[6, 18, 1, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -7, 3, -8, 4, -10}, > {2, -1, 9, -3, 5, -4, 6, -9, 7, -2, 10, -6, 8, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 5 11 16 21 22 21 -q + ----- - ---- + ---- - ---- + ---- - ------- + 16 Sqrt[q] - 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 3/2 5/2 7/2 > 11 q + 5 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 2 -16 4 3 4 -8 -6 5 3 2 4 6 + q - --- - q + --- - --- + --- + q - q + -- - -- - 2 q - q + 18 14 12 10 4 2 q q q q q q 6 8 10 > 3 q - 3 q + q
In[8]:=	HOMFLYPT[Link[10, Alternating, 56]][a, z]
Out[8]=	3 3 5 a a z 3 3 3 5 3 z 5 3 5 7 -(-) + -- - a z - -- + 3 a z - 3 a z + a z - -- + 3 a z - 2 a z + a z z z a a
In[9]:=	Kauffman[Link[10, Alternating, 56]][a, z]
Out[9]=	3 3 4 2 a a 3 6 z 3 3 3 5 3 4 4 z -a + - + -- - a z - a z - ---- - 18 a z - 18 a z - 6 a z - 4 z + ---- - z z a 2 a 5 5 2 4 4 4 6 4 z 16 z 5 3 5 5 5 > 16 a z - 4 a z + 4 a z - -- + ----- + 39 a z + 39 a z + 16 a z - 3 a a 6 7 7 5 6 5 z 2 6 4 6 6 6 11 z 7 > a z + 16 z - ---- + 42 a z + 16 a z - 5 a z - ----- - 14 a z - 2 a a 3 7 5 7 8 2 8 4 8 9 3 9 > 14 a z - 11 a z - 12 z - 24 a z - 12 a z - 5 a z - 5 a z
In[10]:=	Kh[L][q, t]
Out[10]=	10 1 4 1 7 4 9 7 12 12 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 5 10 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 10 11 11 2 2 2 4 2 4 3 6 3 > ----- + ---- + ---- + 7 t + 9 q t + 4 q t + 7 q t + q t + 4 q t + 4 2 4 2 q t q t q t 8 4 > q t

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L10a55 L10a57