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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X4,15,1,16 X5,12,6,13 X8493 X9,16,10,17 X11,20,12,21 X17,22,18,5 X21,18,22,19 X19,10,20,11 X2,14,3,13

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

- q-32 + q-30 - q-28 - 3q-26 - 3q-22 + q-18 + q-16 + 4q-14 + q-12 + 4q-10 + 3q-8 + 2q-4

HOMFLY-PT Polynomial:

- 2a3z-1 - 4a3z - 2a3z3 + 2a5z-1 + 3a5z + 2a5z3 + a5z5 + a7z-1 + 2a7z + 2a7z3 + a7z5 - a9z-1 - a9z - a9z3

Kauffman Polynomial:

- 2a3z-1 + 5a3z - 3a3z3 + 3a4 - 3a4z2 - a4z6 - 2a5z-1 + 6a5z - 8a5z3 + 3a5z5 - 2a5z7 + a6z2 - 3a6z4 + 2a6z6 - 2a6z8 + a7z-1 + 2a7z - 11a7z3 + 11a7z5 - 3a7z7 - a7z9 - 3a8 + 2a8z2 - 3a8z4 + 10a8z6 - 5a8z8 + a9z-1 + 2a9z - 12a9z3 + 17a9z5 - 4a9z7 - a9z9 - 5a10z2 + 3a10z4 + 6a10z6 - 3a10z8 + a11z - 6a11z3 + 9a11z5 - 3a11z7 + a12 - 3a12z2 + 3a12z4 - a12z6

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -2                   2 j = -4                 3 2 j = -6               4     j = -8             4 3     j = -10           6 4       j = -12         3 4         j = -14       4 6           j = -16     2 3             j = -18   1 4               j = -20   2                 j = -22 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[11, NonAlternating, 18]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 18]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[4, 15, 1, 16], X[5, 12, 6, 13], > X[8, 4, 9, 3], X[9, 16, 10, 17], X[11, 20, 12, 21], X[17, 22, 18, 5], > X[21, 18, 22, 19], X[19, 10, 20, 11], X[2, 14, 3, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {-4, -1, 2, -5, -6, 10, -7, 4, 11, -2, 3, 6, -8, 9, > -10, 7, -9, 8}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 3 6 7 9 10 8 7 3 2 q - ----- + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 -30 -28 3 3 -18 -16 4 -12 4 3 2 -q + q - q - --- - --- + q + q + --- + q + --- + -- + -- 26 22 14 10 8 4 q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 18]][a, z]
Out[8]=	3 5 7 9 -2 a 2 a a a 3 5 7 9 3 3 5 3 ----- + ---- + -- - -- - 4 a z + 3 a z + 2 a z - a z - 2 a z + 2 a z + z z z z 7 3 9 3 5 5 7 5 > 2 a z - a z + a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 18]][a, z]
Out[9]=	3 5 7 9 4 8 12 2 a 2 a a a 3 5 7 9 3 a - 3 a + a - ---- - ---- + -- + -- + 5 a z + 6 a z + 2 a z + 2 a z + z z z z 11 4 2 6 2 8 2 10 2 12 2 3 3 > a z - 3 a z + a z + 2 a z - 5 a z - 3 a z - 3 a z - 5 3 7 3 9 3 11 3 6 4 8 4 10 4 > 8 a z - 11 a z - 12 a z - 6 a z - 3 a z - 3 a z + 3 a z + 12 4 5 5 7 5 9 5 11 5 4 6 6 6 > 3 a z + 3 a z + 11 a z + 17 a z + 9 a z - a z + 2 a z + 8 6 10 6 12 6 5 7 7 7 9 7 11 7 > 10 a z + 6 a z - a z - 2 a z - 3 a z - 4 a z - 3 a z - 6 8 8 8 10 8 7 9 9 9 > 2 a z - 5 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 1 2 1 4 2 3 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 22 9 20 8 18 8 18 7 16 7 16 6 14 6 q q q t q t q t q t q t q t q t 6 3 4 6 4 4 3 4 3 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- 14 5 12 5 12 4 10 4 10 3 8 3 8 2 6 2 4 q t q t q t q t q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n18

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