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

K11a227 K11a229

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 18 22 20 14 2 8 6 10 16

Alexander Polynomial:

- t-3 + 10t-2 - 32t-1 + 47 - 32t + 10t2 - t3

Conway Polynomial:

1 - z2 + 4z4 - z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{133, 0}

Jones Polynomial:

q-6 - 3q-5 + 7q-4 - 13q-3 + 18q-2 - 21q-1 + 22 - 19q + 15q2 - 9q3 + 4q4 - q5

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

{K11a251, K11a253, ...}

A2 (sl(3)) Invariant:

q-20 + q-18 - 2q-16 + q-14 + q-12 - 5q-10 + 3q-8 - q-6 - q-4 + 4q-2 - 2 + 5q2 - 3q4 + 3q8 - 4q10 + 2q12 + q14 - q16

HOMFLY-PT Polynomial:

- a-4z2 + a-2z2 + 2a-2z4 + 1 - z2 - z4 - z6 + a2 + 3a2z2 + 3a2z4 - 2a4 - 3a4z2 + a6

Kauffman Polynomial:

- a-5z3 + a-5z5 + a-4z2 - 5a-4z4 + 4a-4z6 + 5a-3z3 - 12a-3z5 + 8a-3z7 - 4a-2z2 + 12a-2z4 - 17a-2z6 + 10a-2z8 + a-1z - a-1z3 - 6a-1z7 + 7a-1z9 + 1 - 7z2 + 22z4 - 29z6 + 12z8 + 2z10 + az - 7az3 + 13az5 - 19az7 + 11az9 - a2 + 3a2z2 + 3a2z4 - 14a2z6 + 6a2z8 + 2a2z10 - 2a3z + 7a3z3 - 8a3z5 - 2a3z7 + 4a3z9 - 2a4 + 8a4z2 - 5a4z4 - 5a4z6 + 4a4z8 - 2a5z + 7a5z3 - 8a5z5 + 3a5z7 - a6 + 3a6z2 - 3a6z4 + a6z6

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

{-1, 2}

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=0 is the signature of 11228. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 11                       1 j = 9                     3   j = 7                   6 1   j = 5                 9 3     j = 3               10 6       j = 1             12 9         j = -1           10 11           j = -3         8 11             j = -5       5 10               j = -7     2 8                 j = -9   1 5                   j = -11   2                     j = -13 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, 228]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 228]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 3, 13, 4], X[18, 5, 19, 6], X[22, 8, 1, 7], > X[20, 9, 21, 10], X[14, 12, 15, 11], X[2, 13, 3, 14], X[8, 16, 9, 15], > X[6, 17, 7, 18], X[10, 19, 11, 20], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 228]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, -9, 4, -8, 5, -10, 6, -2, 7, -6, 8, -11, 9, -3, 10, > -5, 11, -4]
In[5]:=	DTCode[Knot[11, Alternating, 228]]
Out[5]=	DTCode[4, 12, 18, 22, 20, 14, 2, 8, 6, 10, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 228]][t]
Out[6]=	-3 10 32 2 3 47 - t + -- - -- - 32 t + 10 t - t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 228]][z]
Out[7]=	2 4 6 1 - z + 4 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 228]}
In[9]:=	{KnotDet[Knot[11, Alternating, 228]], KnotSignature[Knot[11, Alternating, 228]]}
Out[9]=	{133, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 228]][q]
Out[10]=	-6 3 7 13 18 21 2 3 4 5 22 + q - -- + -- - -- + -- - -- - 19 q + 15 q - 9 q + 4 q - q 5 4 3 2 q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 228], Knot[11, Alternating, 251], > Knot[11, Alternating, 253]}
In[12]:=	A2Invariant[Knot[11, Alternating, 228]][q]
Out[12]=	-20 -18 2 -14 -12 5 3 -6 -4 4 2 -2 + q + q - --- + q + q - --- + -- - q - q + -- + 5 q - 16 10 8 2 q q q q 4 8 10 12 14 16 > 3 q + 3 q - 4 q + 2 q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 228]][a, z]
Out[13]=	2 2 4 2 4 6 2 z z 2 2 4 2 4 2 z 2 4 6 1 + a - 2 a + a - z - -- + -- + 3 a z - 3 a z - z + ---- + 3 a z - z 4 2 2 a a a
In[14]:=	Kauffman[Knot[11, Alternating, 228]][a, z]
Out[14]=	2 2 2 4 6 z 3 5 2 z 4 z 2 2 1 - a - 2 a - a + - + a z - 2 a z - 2 a z - 7 z + -- - ---- + 3 a z + a 4 2 a a 3 3 3 4 2 6 2 z 5 z z 3 3 3 5 3 4 > 8 a z + 3 a z - -- + ---- - -- - 7 a z + 7 a z + 7 a z + 22 z - 5 3 a a a 4 4 5 5 5 z 12 z 2 4 4 4 6 4 z 12 z 5 > ---- + ----- + 3 a z - 5 a z - 3 a z + -- - ----- + 13 a z - 4 2 5 3 a a a a 6 6 3 5 5 5 6 4 z 17 z 2 6 4 6 6 6 > 8 a z - 8 a z - 29 z + ---- - ----- - 14 a z - 5 a z + a z + 4 2 a a 7 7 8 8 z 6 z 7 3 7 5 7 8 10 z 2 8 > ---- - ---- - 19 a z - 2 a z + 3 a z + 12 z + ----- + 6 a z + 3 a 2 a a 9 4 8 7 z 9 3 9 10 2 10 > 4 a z + ---- + 11 a z + 4 a z + 2 z + 2 a z a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 228]], Vassiliev[3][Knot[11, Alternating, 228]]}
Out[15]=	{-1, 2}
In[16]:=	Kh[Knot[11, Alternating, 228]][q, t]
Out[16]=	11 1 2 1 5 2 8 5 10 -- + 12 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 8 11 10 3 3 2 5 2 5 3 > ----- + ---- + --- + 9 q t + 10 q t + 6 q t + 9 q t + 3 q t + 3 2 3 q t q t q t 7 3 7 4 9 4 11 5 > 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a228

K11a227 K11a229