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

K11a182 K11a184

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 14 16 18 20 2 22 10 8 6

Alexander Polynomial:

2t-3 - 11t-2 + 27t-1 - 35 + 27t - 11t2 + 2t3

Conway Polynomial:

1 + z2 + z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{10121, K11a41, K11a198, K11a331, ...}

Determinant and Signature:

{115, -2}

Jones Polynomial:

q-9 - 3q-8 + 6q-7 - 11q-6 + 15q-5 - 18q-4 + 19q-3 - 16q-2 + 13q-1 - 8 + 4q - q2

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

{K11a41, ...}

A2 (sl(3)) Invariant:

q-28 - q-24 + 2q-22 - 3q-20 - 2q-18 + 2q-16 - 3q-14 + 3q-12 + q-8 + 3q-6 - 3q-4 + 4q-2 - 1 - q2 + 2q4 - q6

HOMFLY-PT Polynomial:

- z2 - z4 + a2 + 2a2z2 + 2a2z4 + a2z6 + 2a4 + 3a4z2 + 2a4z4 + a4z6 - 3a6 - 4a6z2 - 2a6z4 + a8 + a8z2

Kauffman Polynomial:

- a-1z3 + a-1z5 + 2z2 - 6z4 + 4z6 + 4az3 - 11az5 + 7az7 - a2 + 2a2z2 - 8a2z6 + 7a2z8 + a3z + 5a3z3 - 12a3z5 + 3a3z7 + 4a3z9 + 2a4 - 7a4z2 + 17a4z4 - 23a4z6 + 11a4z8 + a4z10 - a5z + 2a5z3 - 8a5z7 + 7a5z9 + 3a6 - 10a6z2 + 19a6z4 - 21a6z6 + 8a6z8 + a6z10 - 4a7z + 10a7z3 - 9a7z5 - a7z7 + 3a7z9 + a8 - a8z2 + 5a8z4 - 9a8z6 + 4a8z8 - 2a9z + 8a9z3 - 9a9z5 + 3a9z7 + 2a10z2 - 3a10z4 + a10z6

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

{1, 0}

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 11183. 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                     3   j = 1                   5 1   j = -1                 8 3     j = -3               9 6       j = -5             10 7         j = -7           8 9           j = -9         7 10             j = -11       4 8               j = -13     2 7                 j = -15   1 4                   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, 183]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 183]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 3, 13, 4], X[14, 6, 15, 5], X[16, 7, 17, 8], > X[18, 9, 19, 10], X[20, 11, 21, 12], X[2, 13, 3, 14], X[22, 16, 1, 15], > X[10, 17, 11, 18], X[8, 19, 9, 20], X[6, 21, 7, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 183]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, -11, 4, -10, 5, -9, 6, -2, 7, -3, 8, -4, 9, -5, 10, > -6, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 183]]
Out[5]=	DTCode[4, 12, 14, 16, 18, 20, 2, 22, 10, 8, 6]
In[6]:=	alex = Alexander[Knot[11, Alternating, 183]][t]
Out[6]=	2 11 27 2 3 -35 + -- - -- + -- + 27 t - 11 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 183]][z]
Out[7]=	2 4 6 1 + z + z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 121], Knot[11, Alternating, 41], Knot[11, Alternating, 183], > Knot[11, Alternating, 198], Knot[11, Alternating, 331]}
In[9]:=	{KnotDet[Knot[11, Alternating, 183]], KnotSignature[Knot[11, Alternating, 183]]}
Out[9]=	{115, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 183]][q]
Out[10]=	-9 3 6 11 15 18 19 16 13 2 -8 + q - -- + -- - -- + -- - -- + -- - -- + -- + 4 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, 41], Knot[11, Alternating, 183]}
In[12]:=	A2Invariant[Knot[11, Alternating, 183]][q]
Out[12]=	-28 -24 2 3 2 2 3 3 -8 3 3 4 -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 + 2 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 183]][a, z]
Out[13]=	2 4 6 8 2 2 2 4 2 6 2 8 2 4 a + 2 a - 3 a + a - z + 2 a z + 3 a z - 4 a z + a z - z + 2 4 4 4 6 4 2 6 4 6 > 2 a z + 2 a z - 2 a z + a z + a z
In[14]:=	Kauffman[Knot[11, Alternating, 183]][a, z]
Out[14]=	2 4 6 8 3 5 7 9 2 2 2 -a + 2 a + 3 a + a + a z - a z - 4 a z - 2 a z + 2 z + 2 a z - 3 4 2 6 2 8 2 10 2 z 3 3 3 5 3 > 7 a z - 10 a z - a z + 2 a z - -- + 4 a z + 5 a z + 2 a z + a 5 7 3 9 3 4 4 4 6 4 8 4 10 4 z > 10 a z + 8 a z - 6 z + 17 a z + 19 a z + 5 a z - 3 a z + -- - a 5 3 5 7 5 9 5 6 2 6 4 6 > 11 a z - 12 a z - 9 a z - 9 a z + 4 z - 8 a z - 23 a z - 6 6 8 6 10 6 7 3 7 5 7 7 7 > 21 a z - 9 a z + a z + 7 a z + 3 a z - 8 a z - a z + 9 7 2 8 4 8 6 8 8 8 3 9 5 9 > 3 a z + 7 a z + 11 a z + 8 a z + 4 a z + 4 a z + 7 a z + 7 9 4 10 6 10 > 3 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 183]], Vassiliev[3][Knot[11, Alternating, 183]]}
Out[15]=	{1, 0}
In[16]:=	Kh[Knot[11, Alternating, 183]][q, t]
Out[16]=	6 8 1 2 1 4 2 7 4 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 8 7 10 8 9 10 7 9 3 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 > 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a183

K11a182 K11a184