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

K11a346 K11a348

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 16 12 20 18 4 22 10 2 8 14

Alexander Polynomial:

2t-3 - 11t-2 + 26t-1 - 33 + 26t - 11t2 + 2t3

Conway Polynomial:

1 + z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{10113, K11a107, ...}

Determinant and Signature:

{111, 2}

Jones Polynomial:

q-3 - 3q-2 + 7q-1 - 11 + 15q - 18q2 + 18q3 - 15q4 + 12q5 - 7q6 + 3q7 - q8

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

{K11a159, ...}

A2 (sl(3)) Invariant:

q-10 - q-6 + 3q-4 - q-2 + 2q2 - 5q4 + q6 - 2q8 + 2q10 + 4q12 - q14 + 4q16 - 2q18 - 2q20 + q22 - q24

HOMFLY-PT Polynomial:

- 2a-6 - 2a-6z2 - a-6z4 + 5a-4 + 6a-4z2 + 3a-4z4 + a-4z6 - 3a-2 - 2a-2z2 + a-2z4 + a-2z6 - 3z2 - 2z4 + a2 + a2z2

Kauffman Polynomial:

- 2a-9z3 + a-9z5 - 5a-8z4 + 3a-8z6 - 2a-7z + 9a-7z3 - 13a-7z5 + 6a-7z7 + 2a-6 - 14a-6z2 + 28a-6z4 - 22a-6z6 + 8a-6z8 - a-5z3 + 13a-5z5 - 13a-5z7 + 6a-5z9 + 5a-4 - 27a-4z2 + 45a-4z4 - 28a-4z6 + 6a-4z8 + 2a-4z10 + 6a-3z - 26a-3z3 + 40a-3z5 - 29a-3z7 + 10a-3z9 + 3a-2 - 12a-2z2 + 13a-2z4 - 11a-2z6 + 2a-2z8 + 2a-2z10 + 4a-1z - 9a-1z3 + 5a-1z5 - 7a-1z7 + 4a-1z9 + 4z2 - 2z4 - 7z6 + 4z8 + 5az3 - 8az5 + 3az7 - a2 + 3a2z2 - 3a2z4 + a2z6

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

{0, 3}

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 11347. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 17                       1 j = 15                     2   j = 13                   5 1   j = 11                 7 2     j = 9               8 5       j = 7             10 7         j = 5           8 8           j = 3         7 10             j = 1       5 9               j = -1     2 6                 j = -3   1 5                   j = -5   2                     j = -7 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, 347]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 347]]
Out[3]=	PD[X[6, 2, 7, 1], X[16, 3, 17, 4], X[12, 6, 13, 5], X[20, 8, 21, 7], > X[18, 10, 19, 9], X[4, 12, 5, 11], X[22, 13, 1, 14], X[10, 16, 11, 15], > X[2, 17, 3, 18], X[8, 20, 9, 19], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 347]]
Out[4]=	GaussCode[1, -9, 2, -6, 3, -1, 4, -10, 5, -8, 6, -3, 7, -11, 8, -2, 9, -5, 10, > -4, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 347]]
Out[5]=	DTCode[6, 16, 12, 20, 18, 4, 22, 10, 2, 8, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 347]][t]
Out[6]=	2 11 26 2 3 -33 + -- - -- + -- + 26 t - 11 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 347]][z]
Out[7]=	4 6 1 + z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 113], Knot[11, Alternating, 107], Knot[11, Alternating, 347]}
In[9]:=	{KnotDet[Knot[11, Alternating, 347]], KnotSignature[Knot[11, Alternating, 347]]}
Out[9]=	{111, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 347]][q]
Out[10]=	-3 3 7 2 3 4 5 6 7 8 -11 + q - -- + - + 15 q - 18 q + 18 q - 15 q + 12 q - 7 q + 3 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 159], Knot[11, Alternating, 347]}
In[12]:=	A2Invariant[Knot[11, Alternating, 347]][q]
Out[12]=	-10 -6 3 -2 2 4 6 8 10 12 14 16 q - q + -- - q + 2 q - 5 q + q - 2 q + 2 q + 4 q - q + 4 q - 4 q 18 20 22 24 > 2 q - 2 q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 347]][a, z]
Out[13]=	2 2 2 4 4 4 -2 5 3 2 2 2 z 6 z 2 z 2 2 4 z 3 z z -- + -- - -- + a - 3 z - ---- + ---- - ---- + a z - 2 z - -- + ---- + -- + 6 4 2 6 4 2 6 4 2 a a a a a a a a a 6 6 z z > -- + -- 4 2 a a
In[14]:=	Kauffman[Knot[11, Alternating, 347]][a, z]
Out[14]=	2 2 2 2 5 3 2 2 z 6 z 4 z 2 14 z 27 z 12 z 2 2 -- + -- + -- - a - --- + --- + --- + 4 z - ----- - ----- - ----- + 3 a z - 6 4 2 7 3 a 6 4 2 a a a a a a a a 3 3 3 3 3 4 4 4 2 z 9 z z 26 z 9 z 3 4 5 z 28 z 45 z > ---- + ---- - -- - ----- - ---- + 5 a z - 2 z - ---- + ----- + ----- + 9 7 5 3 a 8 6 4 a a a a a a a 4 5 5 5 5 5 13 z 2 4 z 13 z 13 z 40 z 5 z 5 6 > ----- - 3 a z + -- - ----- + ----- + ----- + ---- - 8 a z - 7 z + 2 9 7 5 3 a a a a a a 6 6 6 6 7 7 7 7 3 z 22 z 28 z 11 z 2 6 6 z 13 z 29 z 7 z > ---- - ----- - ----- - ----- + a z + ---- - ----- - ----- - ---- + 8 6 4 2 7 5 3 a a a a a a a a 8 8 8 9 9 9 10 10 7 8 8 z 6 z 2 z 6 z 10 z 4 z 2 z 2 z > 3 a z + 4 z + ---- + ---- + ---- + ---- + ----- + ---- + ----- + ----- 6 4 2 5 3 a 4 2 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 347]], Vassiliev[3][Knot[11, Alternating, 347]]}
Out[15]=	{0, 3}
In[16]:=	Kh[Knot[11, Alternating, 347]][q, t]
Out[16]=	3 1 2 1 5 2 6 5 q 3 9 q + 7 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 10 q t + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 > 8 q t + 8 q t + 10 q t + 7 q t + 8 q t + 5 q t + 7 q t + 11 5 13 5 13 6 15 6 17 7 > 2 q t + 5 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a347

K11a346 K11a348