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

K11a343 K11a345

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X14,4,15,3 X18,6,19,5 X16,7,17,8 X20,10,21,9 X22,12,1,11 X4,14,5,13 X8,15,9,16 X2,18,3,17 X12,20,13,19 X10,22,11,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 14 18 16 20 22 4 8 2 12 10

Alexander Polynomial:

- 3t-3 + 15t-2 - 29t-1 + 35 - 29t + 15t2 - 3t3

Conway Polynomial:

1 + 4z2 - 3z4 - 3z6

Other knots with the same Alexander/Conway Polynomial:

{K11a275, ...}

Determinant and Signature:

{129, 4}

Jones Polynomial:

1 - 3q + 7q2 - 12q3 + 18q4 - 20q5 + 21q6 - 19q7 + 14q8 - 9q9 + 4q10 - q11

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

{...}

A2 (sl(3)) Invariant:

1 - q2 + 2q6 - 3q8 + 5q10 + 3q16 - 4q18 + 3q20 - 3q22 + 2q26 - 3q28 + 2q30 - q34

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- a-13z3 + a-13z5 + a-12z2 - 5a-12z4 + 4a-12z6 - 2a-11z + 7a-11z3 - 13a-11z5 + 8a-11z7 + a-10 - 3a-10z2 + 6a-10z4 - 13a-10z6 + 9a-10z8 + 9a-9z3 - 12a-9z5 - a-9z7 + 6a-9z9 + 2a-8 - 13a-8z2 + 29a-8z4 - 30a-8z6 + 11a-8z8 + 2a-8z10 + 2a-7z - 8a-7z3 + 18a-7z5 - 22a-7z7 + 11a-7z9 + 3a-6 - 20a-6z2 + 34a-6z4 - 27a-6z6 + 7a-6z8 + 2a-6z10 - 4a-5z3 + 8a-5z5 - 10a-5z7 + 5a-5z9 + 3a-4 - 9a-4z2 + 13a-4z4 - 13a-4z6 + 5a-4z8 + 5a-3z3 - 8a-3z5 + 3a-3z7 + 2a-2z2 - 3a-2z4 + a-2z6

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=4 is the signature of 11344. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 j = 23                       1 j = 21                     3   j = 19                   6 1   j = 17                 8 3     j = 15               11 6       j = 13             10 8         j = 11           10 11           j = 9         8 10             j = 7       4 10               j = 5     3 8                 j = 3   1 5                   j = 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, 344]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 344]]
Out[3]=	PD[X[6, 2, 7, 1], X[14, 4, 15, 3], X[18, 6, 19, 5], X[16, 7, 17, 8], > X[20, 10, 21, 9], X[22, 12, 1, 11], X[4, 14, 5, 13], X[8, 15, 9, 16], > X[2, 18, 3, 17], X[12, 20, 13, 19], X[10, 22, 11, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 344]]
Out[4]=	GaussCode[1, -9, 2, -7, 3, -1, 4, -8, 5, -11, 6, -10, 7, -2, 8, -4, 9, -3, 10, > -5, 11, -6]
In[5]:=	DTCode[Knot[11, Alternating, 344]]
Out[5]=	DTCode[6, 14, 18, 16, 20, 22, 4, 8, 2, 12, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 344]][t]
Out[6]=	3 15 29 2 3 35 - -- + -- - -- - 29 t + 15 t - 3 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 344]][z]
Out[7]=	2 4 6 1 + 4 z - 3 z - 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 275], Knot[11, Alternating, 344]}
In[9]:=	{KnotDet[Knot[11, Alternating, 344]], KnotSignature[Knot[11, Alternating, 344]]}
Out[9]=	{129, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 344]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 1 - 3 q + 7 q - 12 q + 18 q - 20 q + 21 q - 19 q + 14 q - 9 q + 4 q - 11 > q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 344]}
In[12]:=	A2Invariant[Knot[11, Alternating, 344]][q]
Out[12]=	2 6 8 10 16 18 20 22 26 28 1 - q + 2 q - 3 q + 5 q + 3 q - 4 q + 3 q - 3 q + 2 q - 3 q + 30 34 > 2 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 344]][a, z]
Out[13]=	2 2 2 2 2 4 4 4 -10 2 3 3 z 6 z 6 z 3 z 2 z 3 z 6 z z -a + -- - -- + -- - --- + ---- - ---- + ---- + ---- + ---- - ---- - -- + 8 6 4 10 8 6 4 2 8 6 4 a a a a a a a a a a a 4 6 6 z 2 z z > -- - ---- - -- 2 6 4 a a a
In[14]:=	Kauffman[Knot[11, Alternating, 344]][a, z]
Out[14]=	2 2 2 2 2 2 -10 2 3 3 2 z 2 z z 3 z 13 z 20 z 9 z 2 z a + -- + -- + -- - --- + --- + --- - ---- - ----- - ----- - ---- + ---- - 8 6 4 11 7 12 10 8 6 4 2 a a a a a a a a a a a 3 3 3 3 3 3 4 4 4 4 z 7 z 9 z 8 z 4 z 5 z 5 z 6 z 29 z 34 z > --- + ---- + ---- - ---- - ---- + ---- - ---- + ---- + ----- + ----- + 13 11 9 7 5 3 12 10 8 6 a a a a a a a a a a 4 4 5 5 5 5 5 5 6 6 13 z 3 z z 13 z 12 z 18 z 8 z 8 z 4 z 13 z > ----- - ---- + --- - ----- - ----- + ----- + ---- - ---- + ---- - ----- - 4 2 13 11 9 7 5 3 12 10 a a a a a a a a a a 6 6 6 6 7 7 7 7 7 8 30 z 27 z 13 z z 8 z z 22 z 10 z 3 z 9 z > ----- - ----- - ----- + -- + ---- - -- - ----- - ----- + ---- + ---- + 8 6 4 2 11 9 7 5 3 10 a a a a a a a a a a 8 8 8 9 9 9 10 10 11 z 7 z 5 z 6 z 11 z 5 z 2 z 2 z > ----- + ---- + ---- + ---- + ----- + ---- + ----- + ----- 8 6 4 9 7 5 8 6 a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 344]], Vassiliev[3][Knot[11, Alternating, 344]]}
Out[15]=	{4, 9}
In[16]:=	Kh[Knot[11, Alternating, 344]][q, t]
Out[16]=	3 3 5 1 2 q q 5 7 7 2 9 2 5 q + 3 q + ---- + --- + -- + 8 q t + 4 q t + 10 q t + 8 q t + 2 t t q t 9 3 11 3 11 4 13 4 13 5 15 5 > 10 q t + 10 q t + 11 q t + 10 q t + 8 q t + 11 q t + 15 6 17 6 17 7 19 7 19 8 21 8 23 9 > 6 q t + 8 q t + 3 q t + 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a344

K11a343 K11a345