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

K11a304 K11a306

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 12 10 22 16 18 20 8 4 2 14

Alexander Polynomial:

- t-4 + 6t-3 - 16t-2 + 28t-1 - 33 + 28t - 16t2 + 6t3 - t4

Conway Polynomial:

1 + 2z2 - 2z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{K11a157, K11a264, ...}

Determinant and Signature:

{135, -2}

Jones Polynomial:

q-7 - 4q-6 + 8q-5 - 14q-4 + 19q-3 - 21q-2 + 22q-1 - 18 + 14q - 9q2 + 4q3 - q4

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

{...}

A2 (sl(3)) Invariant:

q-20 - 2q-18 + 2q-16 - 3q-14 - 2q-12 + 3q-10 - 3q-8 + 6q-6 - q-4 + 2q-2 + 2 - 3q2 + 3q4 - 2q6 + q10 - q12

HOMFLY-PT Polynomial:

- a-2 - 2a-2z2 - a-2z4 + 2 + 7z2 + 7z4 + 2z6 + a2 - 5a2z2 - 9a2z4 - 5a2z6 - a2z8 - a4 + 2a4z2 + 3a4z4 + a4z6

Kauffman Polynomial:

- a-3z + 3a-3z3 - 3a-3z5 + a-3z7 + a-2 - 6a-2z2 + 13a-2z4 - 13a-2z6 + 4a-2z8 - 2a-1z + 3a-1z3 + 9a-1z5 - 17a-1z7 + 6a-1z9 + 2 - 19z2 + 50z4 - 42z6 + 5z8 + 3z10 - 2az + 3az3 + 26az5 - 44az7 + 16az9 - a2 - 19a2z2 + 64a2z4 - 64a2z6 + 15a2z8 + 3a2z10 + 9a3z3 - 5a3z5 - 14a3z7 + 10a3z9 - a4 - 6a4z2 + 20a4z4 - 27a4z6 + 14a4z8 + a5z + 4a5z3 - 15a5z5 + 12a5z7 - 6a6z4 + 8a6z6 - 2a7z3 + 4a7z5 + a8z4

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

{2, -1}

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 11305. 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 = 9                       1 j = 7                     3   j = 5                   6 1   j = 3                 8 3     j = 1               10 6       j = -1             12 8         j = -3           10 11           j = -5         9 11             j = -7       5 10               j = -9     3 9                 j = -11   1 5                   j = -13   3                     j = -15 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, 305]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 305]]
Out[3]=	PD[X[6, 2, 7, 1], X[12, 4, 13, 3], X[10, 5, 11, 6], X[22, 8, 1, 7], > X[16, 9, 17, 10], X[18, 12, 19, 11], X[20, 13, 21, 14], X[8, 15, 9, 16], > X[4, 18, 5, 17], X[2, 19, 3, 20], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 305]]
Out[4]=	GaussCode[1, -10, 2, -9, 3, -1, 4, -8, 5, -3, 6, -2, 7, -11, 8, -5, 9, -6, 10, > -7, 11, -4]
In[5]:=	DTCode[Knot[11, Alternating, 305]]
Out[5]=	DTCode[6, 12, 10, 22, 16, 18, 20, 8, 4, 2, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 305]][t]
Out[6]=	-4 6 16 28 2 3 4 -33 - t + -- - -- + -- + 28 t - 16 t + 6 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 305]][z]
Out[7]=	2 6 8 1 + 2 z - 2 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 157], Knot[11, Alternating, 264], > Knot[11, Alternating, 305]}
In[9]:=	{KnotDet[Knot[11, Alternating, 305]], KnotSignature[Knot[11, Alternating, 305]]}
Out[9]=	{135, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 305]][q]
Out[10]=	-7 4 8 14 19 21 22 2 3 4 -18 + q - -- + -- - -- + -- - -- + -- + 14 q - 9 q + 4 q - q 6 5 4 3 2 q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 305]}
In[12]:=	A2Invariant[Knot[11, Alternating, 305]][q]
Out[12]=	-20 2 2 3 2 3 3 6 -4 2 2 4 2 + q - --- + --- - --- - --- + --- - -- + -- - q + -- - 3 q + 3 q - 18 16 14 12 10 8 6 2 q q q q q q q q 6 10 12 > 2 q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 305]][a, z]
Out[13]=	2 4 -2 2 4 2 2 z 2 2 4 2 4 z 2 4 2 - a + a - a + 7 z - ---- - 5 a z + 2 a z + 7 z - -- - 9 a z + 2 2 a a 4 4 6 2 6 4 6 2 8 > 3 a z + 2 z - 5 a z + a z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 305]][a, z]
Out[14]=	2 -2 2 4 z 2 z 5 2 6 z 2 2 2 + a - a - a - -- - --- - 2 a z + a z - 19 z - ---- - 19 a z - 3 a 2 a a 3 3 4 2 3 z 3 z 3 3 3 5 3 7 3 4 > 6 a z + ---- + ---- + 3 a z + 9 a z + 4 a z - 2 a z + 50 z + 3 a a 4 5 5 13 z 2 4 4 4 6 4 8 4 3 z 9 z 5 > ----- + 64 a z + 20 a z - 6 a z + a z - ---- + ---- + 26 a z - 2 3 a a a 6 3 5 5 5 7 5 6 13 z 2 6 4 6 > 5 a z - 15 a z + 4 a z - 42 z - ----- - 64 a z - 27 a z + 2 a 7 7 8 6 6 z 17 z 7 3 7 5 7 8 4 z > 8 a z + -- - ----- - 44 a z - 14 a z + 12 a z + 5 z + ---- + 3 a 2 a a 9 2 8 4 8 6 z 9 3 9 10 2 10 > 15 a z + 14 a z + ---- + 16 a z + 10 a z + 3 z + 3 a z a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 305]], Vassiliev[3][Knot[11, Alternating, 305]]}
Out[15]=	{2, -1}
In[16]:=	Kh[Knot[11, Alternating, 305]][q, t]
Out[16]=	11 12 1 3 1 5 3 9 5 10 -- + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 3 q 15 6 13 5 11 5 11 4 9 4 9 3 7 3 7 2 q q t q t q t q t q t q t q t q t 9 11 10 8 t 2 3 2 3 3 5 3 > ----- + ---- + ---- + --- + 10 q t + 6 q t + 8 q t + 3 q t + 6 q t + 5 2 5 3 q q t q t q t 5 4 7 4 9 5 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a305

K11a304 K11a306