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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X18,7,19,8 X4,19,1,20 X5,14,6,15 X8493 X9,16,10,17 X15,10,16,11 X11,20,12,21 X13,22,14,5 X21,12,22,13 X2,18,3,17

Gauss Code:

{{1, -11, 5, -3}, {-4, -1, 2, -5, -6, 7, -8, 10, -9, 4, -7, 6, 11, -2, 3, 8, -10, 9}}

Jones Polynomial:

2q-19/2 - 3q-17/2 + 5q-15/2 - 6q-13/2 + 6q-11/2 - 7q-9/2 + 4q-7/2 - 4q-5/2 + 2q-3/2 - q-1/2

A2 (sl(3)) Invariant:

- q-30 - 3q-28 - 2q-26 - 3q-24 + q-20 + 2q-18 + 5q-16 + 2q-14 + 4q-12 + q-10 + q-8 + q-6 + q-2

HOMFLY-PT Polynomial:

- a3z-1 - 4a3z - 4a3z3 - a3z5 - a5z-1 + 2a5z + 7a5z3 + 5a5z5 + a5z7 + 4a7z-1 + 3a7z - 2a7z3 - a7z5 - 2a9z-1 - a9z

Kauffman Polynomial:

- a3z-1 + 5a3z - 8a3z3 + 5a3z5 - a3z7 + a4 - 10a4z4 + 9a4z6 - 2a4z8 + a5z-1 + 3a5z - 14a5z3 + 9a5z5 + a5z7 - a5z9 - 5a6 + 12a6z2 - 17a6z4 + 15a6z6 - 4a6z8 + 4a7z-1 - 5a7z - a7z3 + 6a7z5 - a7z9 - 6a8 + 12a8z2 - 5a8z4 + 4a8z6 - 2a8z8 + 2a9z-1 - 4a9z + 4a9z3 + a9z5 - 2a9z7 + a10 - 3a10z2 + 2a10z4 - 2a10z6 - a11z - a11z3 - a11z5 + 2a12 - 3a12z2

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 0                   1 j = -2                 1   j = -4               3 1   j = -6             3 3     j = -8           4 1       j = -10         2 3         j = -12       4 4           j = -14     1 2             j = -16   2 4               j = -18   1                 j = -20 2

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[11, NonAlternating, 49]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 49]]
Out[4]=	PD[X[6, 1, 7, 2], X[18, 7, 19, 8], X[4, 19, 1, 20], X[5, 14, 6, 15], > X[8, 4, 9, 3], X[9, 16, 10, 17], X[15, 10, 16, 11], X[11, 20, 12, 21], > X[13, 22, 14, 5], X[21, 12, 22, 13], X[2, 18, 3, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {-4, -1, 2, -5, -6, 7, -8, 10, -9, 4, -7, 6, 11, -2, > 3, 8, -10, 9}]
In[6]:=	Jones[L][q]
Out[6]=	2 3 5 6 6 7 4 4 2 1 ----- - ----- + ----- - ----- + ----- - ---- + ---- - ---- + ---- - ------- 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 3 2 3 -20 2 5 2 4 -10 -8 -6 -2 -q - --- - --- - --- + q + --- + --- + --- + --- + q + q + q + q 28 26 24 18 16 14 12 q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 49]][a, z]
Out[8]=	3 5 7 9 a a 4 a 2 a 3 5 7 9 3 3 -(--) - -- + ---- - ---- - 4 a z + 2 a z + 3 a z - a z - 4 a z + z z z z 5 3 7 3 3 5 5 5 7 5 5 7 > 7 a z - 2 a z - a z + 5 a z - a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 49]][a, z]
Out[9]=	3 5 7 9 4 6 8 10 12 a a 4 a 2 a 3 5 a - 5 a - 6 a + a + 2 a - -- + -- + ---- + ---- + 5 a z + 3 a z - z z z z 7 9 11 6 2 8 2 10 2 12 2 > 5 a z - 4 a z - a z + 12 a z + 12 a z - 3 a z - 3 a z - 3 3 5 3 7 3 9 3 11 3 4 4 6 4 > 8 a z - 14 a z - a z + 4 a z - a z - 10 a z - 17 a z - 8 4 10 4 3 5 5 5 7 5 9 5 11 5 > 5 a z + 2 a z + 5 a z + 9 a z + 6 a z + a z - a z + 4 6 6 6 8 6 10 6 3 7 5 7 9 7 > 9 a z + 15 a z + 4 a z - 2 a z - a z + a z - 2 a z - 4 8 6 8 8 8 5 9 7 9 > 2 a z - 4 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 2 1 2 4 1 2 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 6 4 20 7 18 6 16 6 16 5 14 5 14 4 12 4 q q q t q t q t q t q t q t q t 4 2 3 4 1 3 t t 2 > ------ + ------ + ------ + ----- + ---- + ---- + -- + -- + t 12 3 10 3 10 2 8 2 8 6 4 2 q t q t q t q t q t q t q q

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