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

K11a53 K11a55

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8394 X14,6,15,5 X16,7,17,8 X2,9,3,10 X18,11,19,12 X20,13,21,14 X22,16,1,15 X12,17,13,18 X10,19,11,20 X6,21,7,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 14 16 2 18 20 22 12 10 6

Alexander Polynomial:

2t-3 - 13t-2 + 33t-1 - 43 + 33t - 13t2 + 2t3

Conway Polynomial:

1 - z2 - z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11a172, ...}

Determinant and Signature:

{139, -2}

Jones Polynomial:

q-9 - 3q-8 + 7q-7 - 13q-6 + 18q-5 - 22q-4 + 23q-3 - 20q-2 + 16q-1 - 10 + 5q - q2

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

{...}

A2 (sl(3)) Invariant:

q-28 - q-24 + 3q-22 - 3q-20 - 2q-18 + 3q-16 - 4q-14 + 3q-12 - q-10 + 3q-6 - 4q-4 + 5q-2 - 1 - q2 + 3q4 - q6

HOMFLY-PT Polynomial:

1 - z4 + a2z4 + a2z6 + a4 + a4z2 + a4z4 + a4z6 - 2a6 - 3a6z2 - 2a6z4 + a8 + a8z2

Kauffman Polynomial:

a-1z5 + 1 - 5z4 + 5z6 + 3az3 - 14az5 + 10az7 - a2z2 + 2a2z4 - 14a2z6 + 11a2z8 + a3z + 5a3z3 - 14a3z5 + 7a3z9 + a4 - 8a4z2 + 28a4z4 - 37a4z6 + 15a4z8 + 2a4z10 + a5z - a5z3 + 11a5z5 - 21a5z7 + 12a5z9 + 2a6 - 14a6z2 + 34a6z4 - 31a6z6 + 9a6z8 + 2a6z10 - 2a7z + 3a7z3 + 2a7z5 - 8a7z7 + 5a7z9 + a8 - 5a8z2 + 10a8z4 - 12a8z6 + 5a8z8 - 2a9z + 6a9z3 - 8a9z5 + 3a9z7 + 2a10z2 - 3a10z4 + a10z6

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

{-1, 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 1154. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 5                       1 j = 3                     4   j = 1                   6 1   j = -1                 10 4     j = -3               11 7       j = -5             12 9         j = -7           10 11           j = -9         8 12             j = -11       5 10               j = -13     2 8                 j = -15   1 5                   j = -17   2                     j = -19 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, 54]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 54]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[14, 6, 15, 5], X[16, 7, 17, 8], > X[2, 9, 3, 10], X[18, 11, 19, 12], X[20, 13, 21, 14], X[22, 16, 1, 15], > X[12, 17, 13, 18], X[10, 19, 11, 20], X[6, 21, 7, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 54]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -11, 4, -2, 5, -10, 6, -9, 7, -3, 8, -4, 9, -6, 10, > -7, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 54]]
Out[5]=	DTCode[4, 8, 14, 16, 2, 18, 20, 22, 12, 10, 6]
In[6]:=	alex = Alexander[Knot[11, Alternating, 54]][t]
Out[6]=	2 13 33 2 3 -43 + -- - -- + -- + 33 t - 13 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 54]][z]
Out[7]=	2 4 6 1 - z - z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 54], Knot[11, Alternating, 172]}
In[9]:=	{KnotDet[Knot[11, Alternating, 54]], KnotSignature[Knot[11, Alternating, 54]]}
Out[9]=	{139, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 54]][q]
Out[10]=	-9 3 7 13 18 22 23 20 16 2 -10 + q - -- + -- - -- + -- - -- + -- - -- + -- + 5 q - q 8 7 6 5 4 3 2 q q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 54]}
In[12]:=	A2Invariant[Knot[11, Alternating, 54]][q]
Out[12]=	-28 -24 3 3 2 3 4 3 -10 3 4 5 -1 + q - q + --- - --- - --- + --- - --- + --- - q + -- - -- + -- - 22 20 18 16 14 12 6 4 2 q q q q q q q q q 2 4 6 > q + 3 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 54]][a, z]
Out[13]=	4 6 8 4 2 6 2 8 2 4 2 4 4 4 6 4 1 + a - 2 a + a + a z - 3 a z + a z - z + a z + a z - 2 a z + 2 6 4 6 > a z + a z
In[14]:=	Kauffman[Knot[11, Alternating, 54]][a, z]
Out[14]=	4 6 8 3 5 7 9 2 2 4 2 1 + a + 2 a + a + a z + a z - 2 a z - 2 a z - a z - 8 a z - 6 2 8 2 10 2 3 3 3 5 3 7 3 > 14 a z - 5 a z + 2 a z + 3 a z + 5 a z - a z + 3 a z + 5 9 3 4 2 4 4 4 6 4 8 4 10 4 z > 6 a z - 5 z + 2 a z + 28 a z + 34 a z + 10 a z - 3 a z + -- - a 5 3 5 5 5 7 5 9 5 6 2 6 > 14 a z - 14 a z + 11 a z + 2 a z - 8 a z + 5 z - 14 a z - 4 6 6 6 8 6 10 6 7 5 7 7 7 > 37 a z - 31 a z - 12 a z + a z + 10 a z - 21 a z - 8 a z + 9 7 2 8 4 8 6 8 8 8 3 9 5 9 > 3 a z + 11 a z + 15 a z + 9 a z + 5 a z + 7 a z + 12 a z + 7 9 4 10 6 10 > 5 a z + 2 a z + 2 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 54]], Vassiliev[3][Knot[11, Alternating, 54]]}
Out[15]=	{-1, 3}
In[16]:=	Kh[Knot[11, Alternating, 54]][q, t]
Out[16]=	7 10 1 2 1 5 2 8 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 19 8 17 7 15 7 15 6 13 6 13 5 11 5 q q t q t q t q t q t q t q t 10 8 12 10 11 12 9 11 4 t > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + --- + 11 4 9 4 9 3 7 3 7 2 5 2 5 3 q q t q t q t q t q t q t q t q t 2 3 2 5 3 > 6 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a54

K11a53 K11a55