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

K11n90 K11n92

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X12,6,13,5 X7,16,8,17 X9,18,10,19 X2,11,3,12 X20,13,21,14 X15,8,16,9 X17,6,18,7 X22,19,1,20 X14,21,15,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 12 -16 -18 2 20 -8 -6 22 14

Alexander Polynomial:

- t-2 + 8t-1 - 13 + 8t - t2

Conway Polynomial:

1 + 4z2 - z4

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{31, -2}

Jones Polynomial:

- q-10 + 2q-9 - 3q-8 + 4q-7 - 5q-6 + 5q-5 - 4q-4 + 4q-3 - 2q-2 + q-1

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

{...}

A2 (sl(3)) Invariant:

- q-32 - q-30 + q-28 + q-24 + q-22 - 2q-20 - q-18 - q-16 + q-14 + 2q-12 + q-10 + 2q-8 - q-4 + q-2

HOMFLY-PT Polynomial:

a2z2 + 3a4 + 3a4z2 - 3a6 - 2a6z2 - a6z4 + 2a8 + 2a8z2 - a10

Kauffman Polynomial:

a2z2 + 2a3z3 + 3a4 - 6a4z2 + 4a4z4 - a5z - a5z5 + a5z7 + 3a6 - 12a6z2 + 13a6z4 - 8a6z6 + 2a6z8 - a7z + 2a7z3 - 2a7z5 - 2a7z7 + a7z9 + 2a8 - 11a8z2 + 23a8z4 - 18a8z6 + 4a8z8 - 3a9z + 11a9z3 - 6a9z5 - 2a9z7 + a9z9 + a10 - 6a10z2 + 14a10z4 - 10a10z6 + 2a10z8 - 3a11z + 7a11z3 - 5a11z5 + a11z7

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

{4, -9}

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 1191. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -1                   1 j = -3                 2 1 j = -5               2     j = -7             2 2     j = -9           3 2       j = -11         2 2         j = -13       2 3           j = -15     1 2             j = -17   1 2               j = -19   1                 j = -21 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, NonAlternating, 91]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 91]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[7, 16, 8, 17], > X[9, 18, 10, 19], X[2, 11, 3, 12], X[20, 13, 21, 14], X[15, 8, 16, 9], > X[17, 6, 18, 7], X[22, 19, 1, 20], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 91]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, 9, -4, 8, -5, -2, 6, -3, 7, -11, -8, 4, -9, 5, 10, > -7, 11, -10]
In[5]:=	DTCode[Knot[11, NonAlternating, 91]]
Out[5]=	DTCode[4, 10, 12, -16, -18, 2, 20, -8, -6, 22, 14]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 91]][t]
Out[6]=	-2 8 2 -13 - t + - + 8 t - t t
In[7]:=	Conway[Knot[11, NonAlternating, 91]][z]
Out[7]=	2 4 1 + 4 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 91]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 91]], KnotSignature[Knot[11, NonAlternating, 91]]}
Out[9]=	{31, -2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 91]][q]
Out[10]=	-10 2 3 4 5 5 4 4 2 1 -q + -- - -- + -- - -- + -- - -- + -- - -- + - 9 8 7 6 5 4 3 2 q q q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 91]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 91]][q]
Out[12]=	-32 -30 -28 -24 -22 2 -18 -16 -14 2 -10 -q - q + q + q + q - --- - q - q + q + --- + q + 20 12 q q 2 -4 -2 > -- - q + q 8 q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 91]][a, z]
Out[13]=	4 6 8 10 2 2 4 2 6 2 8 2 6 4 3 a - 3 a + 2 a - a + a z + 3 a z - 2 a z + 2 a z - a z
In[14]:=	Kauffman[Knot[11, NonAlternating, 91]][a, z]
Out[14]=	4 6 8 10 5 7 9 11 2 2 4 2 3 a + 3 a + 2 a + a - a z - a z - 3 a z - 3 a z + a z - 6 a z - 6 2 8 2 10 2 3 3 7 3 9 3 11 3 > 12 a z - 11 a z - 6 a z + 2 a z + 2 a z + 11 a z + 7 a z + 4 4 6 4 8 4 10 4 5 5 7 5 9 5 > 4 a z + 13 a z + 23 a z + 14 a z - a z - 2 a z - 6 a z - 11 5 6 6 8 6 10 6 5 7 7 7 9 7 > 5 a z - 8 a z - 18 a z - 10 a z + a z - 2 a z - 2 a z + 11 7 6 8 8 8 10 8 7 9 9 9 > a z + 2 a z + 4 a z + 2 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 91]], Vassiliev[3][Knot[11, NonAlternating, 91]]}
Out[15]=	{4, -9}
In[16]:=	Kh[Knot[11, NonAlternating, 91]][q, t]
Out[16]=	-3 1 1 1 1 2 1 2 2 q + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + q 21 9 19 8 17 8 17 7 15 7 15 6 13 6 q t q t q t q t q t q t q t 3 2 2 3 2 2 2 2 2 > ------ + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- 13 5 11 5 11 4 9 4 9 3 7 3 7 2 5 2 3 q t q t q t q t q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n91

K11n90 K11n92