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

K11a344 K11a346

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X14,4,15,3 X18,6,19,5 X16,7,17,8 X22,10,1,9 X20,12,21,11 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, -6, 11, -5}

DT (Dowker-Thistlethwaite) Code:

6 14 18 16 22 20 4 8 2 12 10

Alexander Polynomial:

- 5t-2 + 23t-1 - 35 + 23t - 5t2

Conway Polynomial:

1 + 3z2 - 5z4

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{91, 2}

Jones Polynomial:

q-1 - 3 + 6q - 9q2 + 13q3 - 14q4 + 14q5 - 12q6 + 9q7 - 6q8 + 3q9 - q10

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

{...}

A2 (sl(3)) Invariant:

q-4 - q-2 - 1 + 3q2 - 2q4 + q6 + 3q8 - q10 + 2q12 - q14 - 3q20 + 3q22 - q26 + 2q28 - q30 - q32

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- a-11z + 4a-11z3 - 4a-11z5 + a-11z7 + a-10 - 5a-10z2 + 13a-10z4 - 12a-10z6 + 3a-10z8 + 2a-9z - 6a-9z3 + 15a-9z5 - 15a-9z7 + 4a-9z9 + 2a-8 - 13a-8z2 + 26a-8z4 - 17a-8z6 - a-8z8 + 2a-8z10 + 4a-7z - 22a-7z3 + 40a-7z5 - 33a-7z7 + 9a-7z9 + 2a-6 - 13a-6z2 + 26a-6z4 - 20a-6z6 + 2a-6z8 + 2a-6z10 + a-5z - 5a-5z3 + 10a-5z5 - 11a-5z7 + 5a-5z9 + a-4 - a-4z2 + 6a-4z4 - 10a-4z6 + 6a-4z8 + 4a-3z3 - 8a-3z5 + 6a-3z7 - a-2 + 3a-2z2 - 6a-2z4 + 5a-2z6 - 3a-1z3 + 3a-1z5 - z2 + z4

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

{3, 7}

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 11345. 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 = 21                       1 j = 19                     2   j = 17                   4 1   j = 15                 5 2     j = 13               7 4       j = 11             7 5         j = 9           7 7           j = 7         6 7             j = 5       3 7               j = 3     3 6                 j = 1   1 4                   j = -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, 345]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 345]]
Out[3]=	PD[X[6, 2, 7, 1], X[14, 4, 15, 3], X[18, 6, 19, 5], X[16, 7, 17, 8], > X[22, 10, 1, 9], X[20, 12, 21, 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, 345]]
Out[4]=	GaussCode[1, -9, 2, -7, 3, -1, 4, -8, 5, -11, 6, -10, 7, -2, 8, -4, 9, -3, 10, > -6, 11, -5]
In[5]:=	DTCode[Knot[11, Alternating, 345]]
Out[5]=	DTCode[6, 14, 18, 16, 22, 20, 4, 8, 2, 12, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 345]][t]
Out[6]=	5 23 2 -35 - -- + -- + 23 t - 5 t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 345]][z]
Out[7]=	2 4 1 + 3 z - 5 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 345]}
In[9]:=	{KnotDet[Knot[11, Alternating, 345]], KnotSignature[Knot[11, Alternating, 345]]}
Out[9]=	{91, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 345]][q]
Out[10]=	1 2 3 4 5 6 7 8 9 10 -3 + - + 6 q - 9 q + 13 q - 14 q + 14 q - 12 q + 9 q - 6 q + 3 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 345]}
In[12]:=	A2Invariant[Knot[11, Alternating, 345]][q]
Out[12]=	-4 -2 2 4 6 8 10 12 14 20 22 -1 + q - q + 3 q - 2 q + q + 3 q - q + 2 q - q - 3 q + 3 q - 26 28 30 32 > q + 2 q - q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 345]][a, z]
Out[13]=	2 2 2 4 4 4 -10 2 2 -4 -2 2 3 z 2 z z 2 z 2 z z -a + -- - -- + a + a + z + ---- - ---- + -- - ---- - ---- - -- 8 6 8 6 2 6 4 2 a a a a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 345]][a, z]
Out[14]=	2 2 2 -10 2 2 -4 -2 z 2 z 4 z z 2 5 z 13 z 13 z a + -- + -- + a - a - --- + --- + --- + -- - z - ---- - ----- - ----- - 8 6 11 9 7 5 10 8 6 a a a a a a a a a 2 2 3 3 3 3 3 3 4 4 z 3 z 4 z 6 z 22 z 5 z 4 z 3 z 4 13 z 26 z > -- + ---- + ---- - ---- - ----- - ---- + ---- - ---- + z + ----- + ----- + 4 2 11 9 7 5 3 a 10 8 a a a a a a a a a 4 4 4 5 5 5 5 5 5 6 26 z 6 z 6 z 4 z 15 z 40 z 10 z 8 z 3 z 12 z > ----- + ---- - ---- - ---- + ----- + ----- + ----- - ---- + ---- - ----- - 6 4 2 11 9 7 5 3 a 10 a a a a a a a a a 6 6 6 6 7 7 7 7 7 8 17 z 20 z 10 z 5 z z 15 z 33 z 11 z 6 z 3 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 z 2 z 6 z 4 z 9 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, 345]], Vassiliev[3][Knot[11, Alternating, 345]]}
Out[15]=	{3, 7}
In[16]:=	Kh[Knot[11, Alternating, 345]][q, t]
Out[16]=	3 1 2 q 3 5 5 2 7 2 7 3 4 q + 3 q + ----- + --- + - + 6 q t + 3 q t + 7 q t + 6 q t + 7 q t + 3 2 q t t q t 9 3 9 4 11 4 11 5 13 5 13 6 15 6 > 7 q t + 7 q t + 7 q t + 5 q t + 7 q t + 4 q t + 5 q t + 15 7 17 7 17 8 19 8 21 9 > 2 q t + 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a345

K11a344 K11a346