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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-25/2 + 3q-23/2 - 6q-21/2 + 11q-19/2 - 13q-17/2 + 14q-15/2 - 14q-13/2 + 10q-11/2 - 8q-9/2 + 3q-7/2 - q-5/2

A2 (sl(3)) Invariant:

q-40 + 2q-38 - q-36 - 2q-34 - q-32 - 6q-30 - q-28 + q-24 + 6q-22 + q-20 + 6q-18 + q-16 + 3q-12 - 2q-10 + q-8

HOMFLY-PT Polynomial:

- a5z - 2a5z3 - a5z5 - 4a7z-1 - 11a7z - 10a7z3 - 3a7z5 + 8a9z-1 + 14a9z + 6a9z3 - 5a11z-1 - 4a11z + a13z-1

Kauffman Polynomial:

- a5z + 2a5z3 - a5z5 - a6z2 + 4a6z4 - 3a6z6 - 4a7z-1 + 12a7z - 15a7z3 + 13a7z5 - 6a7z7 + 8a8 - 19a8z2 + 18a8z4 - 2a8z6 - 4a8z8 - 8a9z-1 + 22a9z - 32a9z3 + 32a9z5 - 13a9z7 - a9z9 + 14a10 - 35a10z2 + 29a10z4 - a10z6 - 7a10z8 - 5a11z-1 + 9a11z - 16a11z3 + 23a11z5 - 11a11z7 - a11z9 + 9a12 - 22a12z2 + 21a12z4 - 5a12z6 - 3a12z8 - a13z-1 - a13z + a13z3 + 4a13z5 - 4a13z7 + 2a14 - 5a14z2 + 6a14z4 - 3a14z6 - a15z + 2a15z3 - a15z5

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 = -4                     1 j = -6                   3 1 j = -8                 5     j = -10               5 3     j = -12             9 5       j = -14           6 6         j = -16         7 8           j = -18       4 6             j = -20     2 7               j = -22   1 4                 j = -24   2                   j = -26 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, 38]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 38]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[16, 13, 17, 14], X[14, 9, 15, 10], > X[10, 15, 11, 16], X[20, 17, 5, 18], X[18, 7, 19, 8], X[8, 19, 9, 20], > X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 7, -8, 4, -5, 10, -2, 3, -4, 5, -3, 6, -7, > 8, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(25/2) 3 6 11 13 14 14 10 8 -q + ----- - ----- + ----- - ----- + ----- - ----- + ----- - ---- + 23/2 21/2 19/2 17/2 15/2 13/2 11/2 9/2 q q q q q q q q 3 -(5/2) > ---- - q 7/2 q
In[7]:=	A2Invariant[L][q]
Out[7]=	-40 2 -36 2 -32 6 -28 -24 6 -20 6 -16 q + --- - q - --- - q - --- - q + q + --- + q + --- + q + 38 34 30 22 18 q q q q q 3 2 -8 > --- - --- + q 12 10 q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 38]][a, z]
Out[8]=	7 9 11 13 -4 a 8 a 5 a a 5 7 9 11 5 3 ----- + ---- - ----- + --- - a z - 11 a z + 14 a z - 4 a z - 2 a z - z z z z 7 3 9 3 5 5 7 5 > 10 a z + 6 a z - a z - 3 a z
In[9]:=	Kauffman[Link[10, Alternating, 38]][a, z]
Out[9]=	7 9 11 13 8 10 12 14 4 a 8 a 5 a a 5 7 8 a + 14 a + 9 a + 2 a - ---- - ---- - ----- - --- - a z + 12 a z + z z z z 9 11 13 15 6 2 8 2 10 2 > 22 a z + 9 a z - a z - a z - a z - 19 a z - 35 a z - 12 2 14 2 5 3 7 3 9 3 11 3 13 3 > 22 a z - 5 a z + 2 a z - 15 a z - 32 a z - 16 a z + a z + 15 3 6 4 8 4 10 4 12 4 14 4 5 5 > 2 a z + 4 a z + 18 a z + 29 a z + 21 a z + 6 a z - a z + 7 5 9 5 11 5 13 5 15 5 6 6 8 6 > 13 a z + 32 a z + 23 a z + 4 a z - a z - 3 a z - 2 a z - 10 6 12 6 14 6 7 7 9 7 11 7 13 7 > a z - 5 a z - 3 a z - 6 a z - 13 a z - 11 a z - 4 a z - 8 8 10 8 12 8 9 9 11 9 > 4 a z - 7 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 -4 1 2 1 4 2 7 4 q + q + ------- + ------ + ------ + ------ + ------ + ------ + ------ + 26 10 24 9 22 9 22 8 20 8 20 7 18 7 q t q t q t q t q t q t q t 6 7 8 6 6 9 5 5 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 18 6 16 6 16 5 14 5 14 4 12 4 12 3 10 3 q t q t q t q t q t q t q t q t 3 5 3 > ------ + ----- + ---- 10 2 8 2 6 q t q t q t

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