© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11a89 K11a91

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X12,6,13,5 X16,7,17,8 X18,9,19,10 X2,11,3,12 X22,14,1,13 X20,16,21,15 X8,17,9,18 X6,19,7,20 X14,22,15,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 12 16 18 2 22 20 8 6 14

Alexander Polynomial:

2t-3 - 10t-2 + 20t-1 - 23 + 20t - 10t2 + 2t3

Conway Polynomial:

1 - 2z2 + 2z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11a118, ...}

Determinant and Signature:

{87, -2}

Jones Polynomial:

q-7 - 3q-6 + 6q-5 - 9q-4 + 12q-3 - 14q-2 + 13q-1 - 11 + 9q - 5q2 + 3q3 - q4

Other knots (up to mirrors) with the same Jones Polynomial:

{...}

A2 (sl(3)) Invariant:

q-22 - q-18 + 2q-16 - q-14 + q-10 - 3q-8 + q-6 - 2q-4 + q-2 + 2 - q2 + 3q4 + q10 - q12

HOMFLY-PT Polynomial:

- 2a-2z2 - a-2z4 + 2 + 3z2 + 3z4 + z6 - a2 + 2a2z4 + a2z6 - a4 - 4a4z2 - 2a4z4 + a6 + a6z2

Kauffman Polynomial:

4a-3z3 - 4a-3z5 + a-3z7 - 7a-2z2 + 17a-2z4 - 13a-2z6 + 3a-2z8 + a-1z - 2a-1z3 + 8a-1z5 - 10a-1z7 + 3a-1z9 + 2 - 15z2 + 30z4 - 24z6 + 4z8 + z10 + az - 3az3 + 10az5 - 16az7 + 6az9 + a2 - 7a2z2 + 18a2z4 - 21a2z6 + 6a2z8 + a2z10 - 3a3z + 14a3z3 - 14a3z5 + a3z7 + 3a3z9 - a4 + 5a4z2 - 2a4z4 - 5a4z6 + 5a4z8 - 3a5z + 8a5z3 - 9a5z5 + 6a5z7 - a6 + 3a6z2 - 6a6z4 + 5a6z6 - 3a7z3 + 3a7z5 - a8z2 + a8z4

V2 and V3, the type 2 and 3 Vassiliev invariants:

{-2, 3}

Khovanov Homology: (The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=-2 is the signature of 1190. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 9                       1 j = 7                     2   j = 5                   3 1   j = 3                 6 2     j = 1               5 3       j = -1             8 6         j = -3           7 6           j = -5         5 7             j = -7       4 7               j = -9     2 5                 j = -11   1 4                   j = -13   2                     j = -15 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]:=	Show[DrawMorseLink[Knot[11, Alternating, 90]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 90]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[16, 7, 17, 8], > X[18, 9, 19, 10], X[2, 11, 3, 12], X[22, 14, 1, 13], X[20, 16, 21, 15], > X[8, 17, 9, 18], X[6, 19, 7, 20], X[14, 22, 15, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 90]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -10, 4, -9, 5, -2, 6, -3, 7, -11, 8, -4, 9, -5, 10, > -8, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 90]]
Out[5]=	DTCode[4, 10, 12, 16, 18, 2, 22, 20, 8, 6, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 90]][t]
Out[6]=	2 10 20 2 3 -23 + -- - -- + -- + 20 t - 10 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 90]][z]
Out[7]=	2 4 6 1 - 2 z + 2 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 90], Knot[11, Alternating, 118]}
In[9]:=	{KnotDet[Knot[11, Alternating, 90]], KnotSignature[Knot[11, Alternating, 90]]}
Out[9]=	{87, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 90]][q]
Out[10]=	-7 3 6 9 12 14 13 2 3 4 -11 + q - -- + -- - -- + -- - -- + -- + 9 q - 5 q + 3 q - q 6 5 4 3 2 q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 90]}
In[12]:=	A2Invariant[Knot[11, Alternating, 90]][q]
Out[12]=	-22 -18 2 -14 -10 3 -6 2 -2 2 4 10 2 + q - q + --- - q + q - -- + q - -- + q - q + 3 q + q - 16 8 4 q q q 12 > q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 90]][a, z]
Out[13]=	2 4 2 4 6 2 2 z 4 2 6 2 4 z 2 4 2 - a - a + a + 3 z - ---- - 4 a z + a z + 3 z - -- + 2 a z - 2 2 a a 4 4 6 2 6 > 2 a z + z + a z
In[14]:=	Kauffman[Knot[11, Alternating, 90]][a, z]
Out[14]=	2 2 4 6 z 3 5 2 7 z 2 2 2 + a - a - a + - + a z - 3 a z - 3 a z - 15 z - ---- - 7 a z + a 2 a 3 3 4 2 6 2 8 2 4 z 2 z 3 3 3 5 3 > 5 a z + 3 a z - a z + ---- - ---- - 3 a z + 14 a z + 8 a z - 3 a a 4 5 7 3 4 17 z 2 4 4 4 6 4 8 4 4 z > 3 a z + 30 z + ----- + 18 a z - 2 a z - 6 a z + a z - ---- + 2 3 a a 5 6 8 z 5 3 5 5 5 7 5 6 13 z 2 6 > ---- + 10 a z - 14 a z - 9 a z + 3 a z - 24 z - ----- - 21 a z - a 2 a 7 7 8 4 6 6 6 z 10 z 7 3 7 5 7 8 3 z > 5 a z + 5 a z + -- - ----- - 16 a z + a z + 6 a z + 4 z + ---- + 3 a 2 a a 9 2 8 4 8 3 z 9 3 9 10 2 10 > 6 a z + 5 a z + ---- + 6 a z + 3 a z + z + a z a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 90]], Vassiliev[3][Knot[11, Alternating, 90]]}
Out[15]=	{-2, 3}
In[16]:=	Kh[Knot[11, Alternating, 90]][q, t]
Out[16]=	6 8 1 2 1 4 2 5 4 7 -- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 3 q 15 6 13 5 11 5 11 4 9 4 9 3 7 3 7 2 q q t q t q t q t q t q t q t q t 5 7 7 6 t 2 3 2 3 3 5 3 > ----- + ---- + ---- + --- + 5 q t + 3 q t + 6 q t + 2 q t + 3 q t + 5 2 5 3 q q t q t q t 5 4 7 4 9 5 > q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a90

K11a89 K11a91