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

K11a241 K11a243

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 14 18 20 22 16 2 12 10 6 8

Alexander Polynomial:

3t-3 - 7t-2 + 9t-1 - 9 + 9t - 7t2 + 3t3

Conway Polynomial:

1 + 8z2 + 11z4 + 3z6

Other knots with the same Alexander/Conway Polynomial:

{K11n93, ...}

Determinant and Signature:

{47, 6}

Jones Polynomial:

q3 - q4 + 3q5 - 4q6 + 6q7 - 7q8 + 7q9 - 7q10 + 5q11 - 3q12 + 2q13 - q14

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

{...}

A2 (sl(3)) Invariant:

q10 + 2q14 + q16 + q18 + q20 - q22 + q24 - q26 - q28 - q32 + q34 - q42

HOMFLY-PT Polynomial:

- a-12 - 3a-12z2 - a-12z4 + a-10 + 4a-10z2 + 4a-10z4 + a-10z6 - 2a-8 + 3a-8z4 + a-8z6 + 3a-6 + 7a-6z2 + 5a-6z4 + a-6z6

Kauffman Polynomial:

a-17z - 3a-17z3 + a-17z5 + 3a-16z2 - 6a-16z4 + 2a-16z6 + 2a-15z3 - 5a-15z5 + 2a-15z7 - a-14z2 + 6a-14z4 - 6a-14z6 + 2a-14z8 + 2a-13z - 10a-13z3 + 17a-13z5 - 9a-13z7 + 2a-13z9 - a-12 + 3a-12z2 - 3a-12z4 + 7a-12z6 - 4a-12z8 + a-12z10 + 2a-11z - 18a-11z3 + 29a-11z5 - 15a-11z7 + 3a-11z9 - a-10 + 5a-10z2 - 13a-10z4 + 12a-10z6 - 5a-10z8 + a-10z10 + a-9z - 3a-9z3 + 3a-9z5 - 3a-9z7 + a-9z9 - 2a-8 + 5a-8z2 - 3a-8z4 - 2a-8z6 + a-8z8 + 2a-7z - 3a-7z5 + a-7z7 - 3a-6 + 7a-6z2 - 5a-6z4 + a-6z6

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

{8, 23}

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=6 is the signature of 11242. 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 = 29                       1 j = 27                     1   j = 25                   2 1   j = 23                 3 1     j = 21               4 2       j = 19             3 3         j = 17           4 4           j = 15         2 3             j = 13       2 4               j = 11     1 2                 j = 9     2                   j = 7 1 1                     j = 5 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, 242]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 242]]
Out[3]=	PD[X[4, 2, 5, 1], X[14, 4, 15, 3], X[18, 6, 19, 5], X[20, 8, 21, 7], > X[22, 10, 1, 9], X[16, 12, 17, 11], X[2, 14, 3, 13], X[12, 16, 13, 15], > X[10, 18, 11, 17], X[6, 20, 7, 19], X[8, 22, 9, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 242]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, -10, 4, -11, 5, -9, 6, -8, 7, -2, 8, -6, 9, -3, 10, > -4, 11, -5]
In[5]:=	DTCode[Knot[11, Alternating, 242]]
Out[5]=	DTCode[4, 14, 18, 20, 22, 16, 2, 12, 10, 6, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 242]][t]
Out[6]=	3 7 9 2 3 -9 + -- - -- + - + 9 t - 7 t + 3 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 242]][z]
Out[7]=	2 4 6 1 + 8 z + 11 z + 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 242], Knot[11, NonAlternating, 93]}
In[9]:=	{KnotDet[Knot[11, Alternating, 242]], KnotSignature[Knot[11, Alternating, 242]]}
Out[9]=	{47, 6}
In[10]:=	J=Jones[Knot[11, Alternating, 242]][q]
Out[10]=	3 4 5 6 7 8 9 10 11 12 13 14 q - q + 3 q - 4 q + 6 q - 7 q + 7 q - 7 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, Alternating, 242]}
In[12]:=	A2Invariant[Knot[11, Alternating, 242]][q]
Out[12]=	10 14 16 18 20 22 24 26 28 32 34 42 q + 2 q + q + q + q - q + q - q - q - q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 242]][a, z]
Out[13]=	2 2 2 4 4 4 4 6 -12 -10 2 3 3 z 4 z 7 z z 4 z 3 z 5 z z -a + a - -- + -- - ---- + ---- + ---- - --- + ---- + ---- + ---- + --- + 8 6 12 10 6 12 10 8 6 10 a a a a a a a a a a 6 6 z z > -- + -- 8 6 a a
In[14]:=	Kauffman[Knot[11, Alternating, 242]][a, z]
Out[14]=	2 2 2 -12 -10 2 3 z 2 z 2 z z 2 z 3 z z 3 z -a - a - -- - -- + --- + --- + --- + -- + --- + ---- - --- + ---- + 8 6 17 13 11 9 7 16 14 12 a a a a a a a a a a 2 2 2 3 3 3 3 3 4 4 5 z 5 z 7 z 3 z 2 z 10 z 18 z 3 z 6 z 6 z > ---- + ---- + ---- - ---- + ---- - ----- - ----- - ---- - ---- + ---- - 10 8 6 17 15 13 11 9 16 14 a a a a a a a a a a 4 4 4 4 5 5 5 5 5 5 3 z 13 z 3 z 5 z z 5 z 17 z 29 z 3 z 3 z > ---- - ----- - ---- - ---- + --- - ---- + ----- + ----- + ---- - ---- + 12 10 8 6 17 15 13 11 9 7 a a a a a a a a a a 6 6 6 6 6 6 7 7 7 7 7 2 z 6 z 7 z 12 z 2 z z 2 z 9 z 15 z 3 z z > ---- - ---- + ---- + ----- - ---- + -- + ---- - ---- - ----- - ---- + -- + 16 14 12 10 8 6 15 13 11 9 7 a a a a a a a a a a a 8 8 8 8 9 9 9 10 10 2 z 4 z 5 z z 2 z 3 z z z z > ---- - ---- - ---- + -- + ---- + ---- + -- + --- + --- 14 12 10 8 13 11 9 12 10 a a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 242]], Vassiliev[3][Knot[11, Alternating, 242]]}
Out[15]=	{8, 23}
In[16]:=	Kh[Knot[11, Alternating, 242]][q, t]
Out[16]=	5 7 7 9 2 11 2 11 3 13 3 13 4 15 4 q + q + q t + 2 q t + q t + 2 q t + 2 q t + 4 q t + 2 q t + 15 5 17 5 17 6 19 6 19 7 21 7 > 3 q t + 4 q t + 4 q t + 3 q t + 3 q t + 4 q t + 21 8 23 8 23 9 25 9 25 10 27 10 29 11 > 2 q t + 3 q t + q t + 2 q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a242

K11a241 K11a243