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

K11a361 K11a363

Knotscape

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DrawMorseLink

PD Presentation:

X8291 X14,4,15,3 X18,6,19,5 X16,8,17,7 X22,10,1,9 X20,12,21,11 X4,14,5,13 X2,16,3,15 X6,18,7,17 X12,20,13,19 X10,22,11,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

8 14 18 16 22 20 4 2 6 12 10

Alexander Polynomial:

10t-1 - 19 + 10t

Conway Polynomial:

1 + 10z2

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{39, 2}

Jones Polynomial:

q - 2q2 + 3q3 - 4q4 + 5q5 - 5q6 + 6q7 - 4q8 + 4q9 - 3q10 + q11 - q12

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

{...}

A2 (sl(3)) Invariant:

q2 - q4 + q8 - q10 + q14 + q18 + q20 + 2q22 + 2q24 + q28 - q30 - 2q32 - q36 - q38

HOMFLY-PT Polynomial:

- a-12 - a-10 + a-10z2 + 2a-8 + 3a-8z2 + a-6 + 3a-6z2 + 2a-4z2 + a-2z2

Kauffman Polynomial:

12a-13z - 28a-13z3 + 23a-13z5 - 8a-13z7 + a-13z9 - a-12 - 3a-12z2 - 2a-12z4 + 10a-12z6 - 6a-12z8 + a-12z10 + 14a-11z - 47a-11z3 + 56a-11z5 - 26a-11z7 + 4a-11z9 + a-10 - 24a-10z2 + 41a-10z4 - 17a-10z6 - a-10z8 + a-10z10 + 2a-9z - 2a-9z3 + 13a-9z5 - 13a-9z7 + 3a-9z9 + 2a-8 - 12a-8z2 + 27a-8z4 - 22a-8z6 + 5a-8z8 + 9a-7z3 - 16a-7z5 + 5a-7z7 - a-6 + 6a-6z2 - 13a-6z4 + 5a-6z6 - 6a-5z3 + 4a-5z5 - 2a-4z2 + 3a-4z4 + 2a-3z3 + a-2z2

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

{10, 31}

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 11362. 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 r = 10 r = 11 j = 25                       1 j = 23                         j = 21                   3 1   j = 19                 1       j = 17               3 3       j = 15             3 1         j = 13           2 3           j = 11         3 3             j = 9       1 2               j = 7     2 3                 j = 5     1                   j = 3 1 2                     j = 1 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, 362]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 362]]
Out[3]=	PD[X[8, 2, 9, 1], X[14, 4, 15, 3], X[18, 6, 19, 5], X[16, 8, 17, 7], > X[22, 10, 1, 9], X[20, 12, 21, 11], X[4, 14, 5, 13], X[2, 16, 3, 15], > X[6, 18, 7, 17], X[12, 20, 13, 19], X[10, 22, 11, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 362]]
Out[4]=	GaussCode[1, -8, 2, -7, 3, -9, 4, -1, 5, -11, 6, -10, 7, -2, 8, -4, 9, -3, 10, > -6, 11, -5]
In[5]:=	DTCode[Knot[11, Alternating, 362]]
Out[5]=	DTCode[8, 14, 18, 16, 22, 20, 4, 2, 6, 12, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 362]][t]
Out[6]=	10 -19 + -- + 10 t t
In[7]:=	Conway[Knot[11, Alternating, 362]][z]
Out[7]=	2 1 + 10 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 362]}
In[9]:=	{KnotDet[Knot[11, Alternating, 362]], KnotSignature[Knot[11, Alternating, 362]]}
Out[9]=	{39, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 362]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 11 12 q - 2 q + 3 q - 4 q + 5 q - 5 q + 6 q - 4 q + 4 q - 3 q + q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 362]}
In[12]:=	A2Invariant[Knot[11, Alternating, 362]][q]
Out[12]=	2 4 8 10 14 18 20 22 24 28 30 32 q - q + q - q + q + q + q + 2 q + 2 q + q - q - 2 q - 36 38 > q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 362]][a, z]
Out[13]=	2 2 2 2 2 -12 -10 2 -6 z 3 z 3 z 2 z z -a - a + -- + a + --- + ---- + ---- + ---- + -- 8 10 8 6 4 2 a a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 362]][a, z]
Out[14]=	2 2 2 2 -12 -10 2 -6 12 z 14 z 2 z 3 z 24 z 12 z 6 z -a + a + -- - a + ---- + ---- + --- - ---- - ----- - ----- + ---- - 8 13 11 9 12 10 8 6 a a a a a a a a 2 2 3 3 3 3 3 3 4 4 2 z z 28 z 47 z 2 z 9 z 6 z 2 z 2 z 41 z > ---- + -- - ----- - ----- - ---- + ---- - ---- + ---- - ---- + ----- + 4 2 13 11 9 7 5 3 12 10 a a a a a a a a a a 4 4 4 5 5 5 5 5 6 27 z 13 z 3 z 23 z 56 z 13 z 16 z 4 z 10 z > ----- - ----- + ---- + ----- + ----- + ----- - ----- + ---- + ----- - 8 6 4 13 11 9 7 5 12 a a a a a a a a a 6 6 6 7 7 7 7 8 8 8 17 z 22 z 5 z 8 z 26 z 13 z 5 z 6 z z 5 z > ----- - ----- + ---- - ---- - ----- - ----- + ---- - ---- - --- + ---- + 10 8 6 13 11 9 7 12 10 8 a a a a a a a a a a 9 9 9 10 10 z 4 z 3 z z z > --- + ---- + ---- + --- + --- 13 11 9 12 10 a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 362]], Vassiliev[3][Knot[11, Alternating, 362]]}
Out[15]=	{10, 31}
In[16]:=	Kh[Knot[11, Alternating, 362]][q, t]
Out[16]=	3 3 5 2 7 2 7 3 9 3 9 4 11 4 q + q + 2 q t + q t + 2 q t + 3 q t + q t + 2 q t + 3 q t + 11 5 13 5 13 6 15 6 15 7 17 7 17 8 > 3 q t + 2 q t + 3 q t + 3 q t + q t + 3 q t + 3 q t + 19 8 21 9 21 10 25 11 > q t + 3 q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a362

K11a361 K11a363