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

K11n62 K11n64

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X5,15,6,14 X2837 X16,9,17,10 X20,12,21,11 X18,14,19,13 X15,7,16,6 X22,17,1,18 X12,20,13,19 X10,22,11,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- 2t-2 + 10t-1 - 15 + 10t - 2t2

Conway Polynomial:

1 + 2z2 - 2z4

Other knots with the same Alexander/Conway Polynomial:

{915, 10165, K11n101, ...}

Determinant and Signature:

{39, 2}

Jones Polynomial:

2q - 3q2 + 5q3 - 6q4 + 6q5 - 6q6 + 5q7 - 3q8 + 2q9 - q10

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

{...}

A2 (sl(3)) Invariant:

2q2 + q6 + 2q8 - q10 - q14 - q16 - q20 + 2q22 + q24 + q28 - q30 - q32

HOMFLY-PT Polynomial:

- a-10 + 2a-8 + 2a-8z2 - a-6 - a-6z2 - a-6z4 - a-4 - a-4z2 - a-4z4 + 2a-2 + 2a-2z2

Kauffman Polynomial:

- 2a-11z + 7a-11z3 - 5a-11z5 + a-11z7 + a-10 - 7a-10z2 + 15a-10z4 - 10a-10z6 + 2a-10z8 - a-9z + 7a-9z3 - 4a-9z5 - 2a-9z7 + a-9z9 + 2a-8 - 12a-8z2 + 20a-8z4 - 16a-8z6 + 4a-8z8 + 3a-7z - 5a-7z3 - a-7z5 - a-7z7 + a-7z9 + a-6 - 5a-6z2 + 3a-6z4 - 4a-6z6 + 2a-6z8 + 3a-5z - 4a-5z3 - a-5z5 + 2a-5z7 - a-4 + 3a-4z2 - 2a-4z4 + 2a-4z6 + a-3z + a-3z3 + a-3z5 - 2a-2 + 3a-2z2

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

{2, 5}

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 1163. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 j = 21                   1 j = 19                 1   j = 17               2 1   j = 15             3 1     j = 13           3 2       j = 11         3 3         j = 9       3 3           j = 7     2 3             j = 5   1 3               j = 3 1 2                 j = 1 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]:=	Show[DrawMorseLink[Knot[11, NonAlternating, 63]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 63]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 15, 6, 14], X[2, 8, 3, 7], > X[16, 9, 17, 10], X[20, 12, 21, 11], X[18, 14, 19, 13], X[15, 7, 16, 6], > X[22, 17, 1, 18], X[12, 20, 13, 19], X[10, 22, 11, 21]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 63]]
Out[4]=	GaussCode[1, -4, 2, -1, -3, 8, 4, -2, 5, -11, 6, -10, 7, 3, -8, -5, 9, -7, 10, > -6, 11, -9]
In[5]:=	DTCode[Knot[11, NonAlternating, 63]]
Out[5]=	DTCode[4, 8, -14, 2, 16, 20, 18, -6, 22, 12, 10]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 63]][t]
Out[6]=	2 10 2 -15 - -- + -- + 10 t - 2 t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 63]][z]
Out[7]=	2 4 1 + 2 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 15], Knot[10, 165], Knot[11, NonAlternating, 63], > Knot[11, NonAlternating, 101]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 63]], KnotSignature[Knot[11, NonAlternating, 63]]}
Out[9]=	{39, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 63]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 2 q - 3 q + 5 q - 6 q + 6 q - 6 q + 5 q - 3 q + 2 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 63]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 63]][q]
Out[12]=	2 6 8 10 14 16 20 22 24 28 30 32 2 q + q + 2 q - q - q - q - q + 2 q + q + q - q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 63]][a, z]
Out[13]=	2 2 2 2 4 4 -10 2 -6 -4 2 2 z z z 2 z z z -a + -- - a - a + -- + ---- - -- - -- + ---- - -- - -- 8 2 8 6 4 2 6 4 a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 63]][a, z]
Out[14]=	2 2 2 -10 2 -6 -4 2 2 z z 3 z 3 z z 7 z 12 z 5 z a + -- + a - a - -- - --- - -- + --- + --- + -- - ---- - ----- - ---- + 8 2 11 9 7 5 3 10 8 6 a a a a a a a a a a 2 2 3 3 3 3 3 4 4 4 3 z 3 z 7 z 7 z 5 z 4 z z 15 z 20 z 3 z > ---- + ---- + ---- + ---- - ---- - ---- + -- + ----- + ----- + ---- - 4 2 11 9 7 5 3 10 8 6 a a a a a a a a a a 4 5 5 5 5 5 6 6 6 6 7 2 z 5 z 4 z z z z 10 z 16 z 4 z 2 z z > ---- - ---- - ---- - -- - -- + -- - ----- - ----- - ---- + ---- + --- - 4 11 9 7 5 3 10 8 6 4 11 a a a a a a a a a a a 7 7 7 8 8 8 9 9 2 z z 2 z 2 z 4 z 2 z z z > ---- - -- + ---- + ---- + ---- + ---- + -- + -- 9 7 5 10 8 6 9 7 a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 63]], Vassiliev[3][Knot[11, NonAlternating, 63]]}
Out[15]=	{2, 5}
In[16]:=	Kh[Knot[11, NonAlternating, 63]][q, t]
Out[16]=	3 3 5 5 2 7 2 7 3 9 3 9 4 2 q + q + 2 q t + q t + 3 q t + 2 q t + 3 q t + 3 q t + 3 q t + 11 4 11 5 13 5 13 6 15 6 15 7 17 7 > 3 q t + 3 q t + 3 q t + 2 q t + 3 q t + q t + 2 q t + 17 8 19 8 21 9 > q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n63

K11n62 K11n64