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

K11a360 K11a362

Knotscape

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DrawMorseLink

PD Presentation:

X8291 X14,4,15,3 X18,6,19,5 X16,8,17,7 X20,10,21,9 X22,12,1,11 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, -5, 11, -6}

DT (Dowker-Thistlethwaite) Code:

8 14 18 16 20 22 4 2 6 12 10

Alexander Polynomial:

7t-2 - 17t-1 + 21 - 17t + 7t2

Conway Polynomial:

1 + 11z2 + 7z4

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{69, 4}

Jones Polynomial:

q2 - 2q3 + 4q4 - 7q5 + 10q6 - 10q7 + 11q8 - 9q9 + 7q10 - 5q11 + 2q12 - q13

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

{...}

A2 (sl(3)) Invariant:

q6 - q8 + q10 - 2q14 + 2q16 + q20 + 3q22 + q24 + 3q26 - q28 - q30 - 3q34 - q40

HOMFLY-PT Polynomial:

- a-12 - a-12z2 - 2a-10 + a-10z4 + 4a-8 + 7a-8z2 + 3a-8z4 + 3a-6z2 + 2a-6z4 + 2a-4z2 + a-4z4

Kauffman Polynomial:

- 4a-15z + 8a-15z3 - 5a-15z5 + a-15z7 - a-14z2 + 8a-14z4 - 8a-14z6 + 2a-14z8 + 3a-13z - 2a-13z3 + 3a-13z5 - 6a-13z7 + 2a-13z9 - a-12 + 2a-12z2 + 3a-12z4 - 7a-12z6 + a-12z10 + 7a-11z - 21a-11z3 + 24a-11z5 - 18a-11z7 + 5a-11z9 + 2a-10 - 21a-10z2 + 32a-10z4 - 20a-10z6 + 3a-10z8 + a-10z10 + a-9z3 + 5a-9z5 - 7a-9z7 + 3a-9z9 + 4a-8 - 19a-8z2 + 31a-8z4 - 18a-8z6 + 5a-8z8 + 9a-7z3 - 9a-7z5 + 4a-7z7 + 3a-6z2 - 5a-6z4 + 3a-6z6 - 3a-5z3 + 2a-5z5 - 2a-4z2 + a-4z4

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

{11, 35}

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=4 is the signature of 11361. 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 = 27                       1 j = 25                     1   j = 23                   4 1   j = 21                 3 1     j = 19               6 4       j = 17             5 3         j = 15           5 6           j = 13         5 5             j = 11       2 5               j = 9     2 5                 j = 7     2                   j = 5 1 2                     j = 3 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, 361]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 361]]
Out[3]=	PD[X[8, 2, 9, 1], X[14, 4, 15, 3], X[18, 6, 19, 5], X[16, 8, 17, 7], > X[20, 10, 21, 9], X[22, 12, 1, 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, 361]]
Out[4]=	GaussCode[1, -8, 2, -7, 3, -9, 4, -1, 5, -11, 6, -10, 7, -2, 8, -4, 9, -3, 10, > -5, 11, -6]
In[5]:=	DTCode[Knot[11, Alternating, 361]]
Out[5]=	DTCode[8, 14, 18, 16, 20, 22, 4, 2, 6, 12, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 361]][t]
Out[6]=	7 17 2 21 + -- - -- - 17 t + 7 t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 361]][z]
Out[7]=	2 4 1 + 11 z + 7 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 361]}
In[9]:=	{KnotDet[Knot[11, Alternating, 361]], KnotSignature[Knot[11, Alternating, 361]]}
Out[9]=	{69, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 361]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 11 q - 2 q + 4 q - 7 q + 10 q - 10 q + 11 q - 9 q + 7 q - 5 q + 12 13 > 2 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 361]}
In[12]:=	A2Invariant[Knot[11, Alternating, 361]][q]
Out[12]=	6 8 10 14 16 20 22 24 26 28 30 34 q - q + q - 2 q + 2 q + q + 3 q + q + 3 q - q - q - 3 q - 40 > q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 361]][a, z]
Out[13]=	2 2 2 2 4 4 4 4 -12 2 4 z 7 z 3 z 2 z z 3 z 2 z z -a - --- + -- - --- + ---- + ---- + ---- + --- + ---- + ---- + -- 10 8 12 8 6 4 10 8 6 4 a a a a a a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 361]][a, z]
Out[14]=	2 2 2 2 2 2 -12 2 4 4 z 3 z 7 z z 2 z 21 z 19 z 3 z 2 z -a + --- + -- - --- + --- + --- - --- + ---- - ----- - ----- + ---- - ---- + 10 8 15 13 11 14 12 10 8 6 4 a a a a a a a a a a a 3 3 3 3 3 3 4 4 4 4 8 z 2 z 21 z z 9 z 3 z 8 z 3 z 32 z 31 z > ---- - ---- - ----- + -- + ---- - ---- + ---- + ---- + ----- + ----- - 15 13 11 9 7 5 14 12 10 8 a a a a a a a a a a 4 4 5 5 5 5 5 5 6 6 5 z z 5 z 3 z 24 z 5 z 9 z 2 z 8 z 7 z > ---- + -- - ---- + ---- + ----- + ---- - ---- + ---- - ---- - ---- - 6 4 15 13 11 9 7 5 14 12 a a a a a a a a a a 6 6 6 7 7 7 7 7 8 8 20 z 18 z 3 z z 6 z 18 z 7 z 4 z 2 z 3 z > ----- - ----- + ---- + --- - ---- - ----- - ---- + ---- + ---- + ---- + 10 8 6 15 13 11 9 7 14 10 a a a a a a a a a a 8 9 9 9 10 10 5 z 2 z 5 z 3 z z z > ---- + ---- + ---- + ---- + --- + --- 8 13 11 9 12 10 a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 361]], Vassiliev[3][Knot[11, Alternating, 361]]}
Out[15]=	{11, 35}
In[16]:=	Kh[Knot[11, Alternating, 361]][q, t]
Out[16]=	3 5 5 7 2 9 2 9 3 11 3 11 4 q + q + 2 q t + 2 q t + 2 q t + 5 q t + 2 q t + 5 q t + 13 4 13 5 15 5 15 6 17 6 17 7 > 5 q t + 5 q t + 5 q t + 6 q t + 5 q t + 3 q t + 19 7 19 8 21 8 21 9 23 9 23 10 25 10 > 6 q t + 4 q t + 3 q t + q t + 4 q t + q t + q t + 27 11 > q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a361

K11a360 K11a362