K11a99

K11a98

K11a100

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a99 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X12,6,13,5 X20,8,21,7 X16,10,17,9 X2,11,3,12 X18,13,19,14 X8,16,9,15 X22,17,1,18 X6,20,7,19 X14,21,15,22
Gauss code	1, -6, 2, -1, 3, -10, 4, -8, 5, -2, 6, -3, 7, -11, 8, -5, 9, -7, 10, -4, 11, -9
Dowker-Thistlethwaite code	4 10 12 20 16 2 18 8 22 6 14

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number [math]\displaystyle{ \{1,2\} }[/math] 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a99/ThurstonBennequinNumber Hyperbolic Volume 16.6086 A-Polynomial See Data:K11a99/A-polynomial

[edit Notes for K11a99's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus [math]\displaystyle{ 4 }[/math] Rasmussen s-Invariant -2

[edit Notes for K11a99's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ -t^4+6 t^3-17 t^2+28 t-31+28 t^{-1} -17 t^{-2} +6 t^{-3} - t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ -z^8-2 z^6-z^4-2 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 135, 2 }
Jones polynomial	[math]\displaystyle{ q^7-4 q^6+9 q^5-14 q^4+19 q^3-22 q^2+21 q-18+14 q^{-1} -8 q^{-2} +4 q^{-3} - q^{-4} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^8 a^{-2} -5 z^6 a^{-2} +z^6 a^{-4} +2 z^6-a^2 z^4-10 z^4 a^{-2} +3 z^4 a^{-4} +7 z^4-2 a^2 z^2-10 z^2 a^{-2} +3 z^2 a^{-4} +7 z^2-4 a^{-2} +2 a^{-4} +3 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ 2 z^{10} a^{-2} +2 z^{10}+5 a z^9+13 z^9 a^{-1} +8 z^9 a^{-3} +4 a^2 z^8+19 z^8 a^{-2} +13 z^8 a^{-4} +10 z^8+a^3 z^7-12 a z^7-27 z^7 a^{-1} -z^7 a^{-3} +13 z^7 a^{-5} -14 a^2 z^6-62 z^6 a^{-2} -18 z^6 a^{-4} +9 z^6 a^{-6} -49 z^6-3 a^3 z^5+a z^5-25 z^5 a^{-3} -17 z^5 a^{-5} +4 z^5 a^{-7} +16 a^2 z^4+55 z^4 a^{-2} +7 z^4 a^{-4} -8 z^4 a^{-6} +z^4 a^{-8} +55 z^4+3 a^3 z^3+10 a z^3+15 z^3 a^{-1} +18 z^3 a^{-3} +9 z^3 a^{-5} -z^3 a^{-7} -6 a^2 z^2-23 z^2 a^{-2} -5 z^2 a^{-4} +3 z^2 a^{-6} -21 z^2-a^3 z-3 a z-3 z a^{-1} -3 z a^{-3} -2 z a^{-5} +4 a^{-2} +2 a^{-4} +3 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{12}+q^{10}+q^8-q^6+4 q^4-2 q^2+2+ q^{-2} -4 q^{-4} +3 q^{-6} -5 q^{-8} +3 q^{-10} - q^{-14} +3 q^{-16} -2 q^{-18} + q^{-20} }[/math]
The G2 invariant	[math]\displaystyle{ q^{60}-3 q^{58}+9 q^{56}-19 q^{54}+28 q^{52}-33 q^{50}+17 q^{48}+26 q^{46}-91 q^{44}+165 q^{42}-204 q^{40}+167 q^{38}-37 q^{36}-174 q^{34}+394 q^{32}-522 q^{30}+480 q^{28}-242 q^{26}-135 q^{24}+516 q^{22}-741 q^{20}+713 q^{18}-405 q^{16}-48 q^{14}+467 q^{12}-683 q^{10}+599 q^8-269 q^6-156 q^4+498 q^2-586+391 q^{-2} +18 q^{-4} -460 q^{-6} +748 q^{-8} -756 q^{-10} +454 q^{-12} +43 q^{-14} -573 q^{-16} +935 q^{-18} -987 q^{-20} +703 q^{-22} -172 q^{-24} -401 q^{-26} +798 q^{-28} -894 q^{-30} +643 q^{-32} -188 q^{-34} -279 q^{-36} +566 q^{-38} -560 q^{-40} +292 q^{-42} +105 q^{-44} -431 q^{-46} +544 q^{-48} -402 q^{-50} +68 q^{-52} +293 q^{-54} -544 q^{-56} +602 q^{-58} -444 q^{-60} +168 q^{-62} +140 q^{-64} -372 q^{-66} +462 q^{-68} -420 q^{-70} +276 q^{-72} -96 q^{-74} -65 q^{-76} +179 q^{-78} -227 q^{-80} +213 q^{-82} -152 q^{-84} +78 q^{-86} -5 q^{-88} -46 q^{-90} +68 q^{-92} -74 q^{-94} +57 q^{-96} -33 q^{-98} +13 q^{-100} +4 q^{-102} -10 q^{-104} +11 q^{-106} -10 q^{-108} +6 q^{-110} -3 q^{-112} + q^{-114} }[/math]

Further Quantum Invariants

K11a99 Quantum Invariants.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11a99"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

[math]\displaystyle{ -t^4+6 t^3-17 t^2+28 t-31+28 t^{-1} -17 t^{-2} +6 t^{-3} - t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ -z^8-2 z^6-z^4-2 z^2+1 }[/math]

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

[math]\displaystyle{ \{1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 135, 2 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

[math]\displaystyle{ q^7-4 q^6+9 q^5-14 q^4+19 q^3-22 q^2+21 q-18+14 q^{-1} -8 q^{-2} +4 q^{-3} - q^{-4} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

[math]\displaystyle{ -z^8 a^{-2} -5 z^6 a^{-2} +z^6 a^{-4} +2 z^6-a^2 z^4-10 z^4 a^{-2} +3 z^4 a^{-4} +7 z^4-2 a^2 z^2-10 z^2 a^{-2} +3 z^2 a^{-4} +7 z^2-4 a^{-2} +2 a^{-4} +3 }[/math]

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

[math]\displaystyle{ 2 z^{10} a^{-2} +2 z^{10}+5 a z^9+13 z^9 a^{-1} +8 z^9 a^{-3} +4 a^2 z^8+19 z^8 a^{-2} +13 z^8 a^{-4} +10 z^8+a^3 z^7-12 a z^7-27 z^7 a^{-1} -z^7 a^{-3} +13 z^7 a^{-5} -14 a^2 z^6-62 z^6 a^{-2} -18 z^6 a^{-4} +9 z^6 a^{-6} -49 z^6-3 a^3 z^5+a z^5-25 z^5 a^{-3} -17 z^5 a^{-5} +4 z^5 a^{-7} +16 a^2 z^4+55 z^4 a^{-2} +7 z^4 a^{-4} -8 z^4 a^{-6} +z^4 a^{-8} +55 z^4+3 a^3 z^3+10 a z^3+15 z^3 a^{-1} +18 z^3 a^{-3} +9 z^3 a^{-5} -z^3 a^{-7} -6 a^2 z^2-23 z^2 a^{-2} -5 z^2 a^{-4} +3 z^2 a^{-6} -21 z^2-a^3 z-3 a z-3 z a^{-1} -3 z a^{-3} -2 z a^{-5} +4 a^{-2} +2 a^{-4} +3 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a277,}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11a99"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]=

{ [math]\displaystyle{ -t^4+6 t^3-17 t^2+28 t-31+28 t^{-1} -17 t^{-2} +6 t^{-3} - t^{-4} }[/math], [math]\displaystyle{ q^7-4 q^6+9 q^5-14 q^4+19 q^3-22 q^2+21 q-18+14 q^{-1} -8 q^{-2} +4 q^{-3} - q^{-4} }[/math] }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{K11a277,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{}

Vassiliev invariants

V2 and V3:

(-2, -2)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 [math]\displaystyle{ -8 }[/math] [math]\displaystyle{ -16 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ \frac{116}{3} }[/math] [math]\displaystyle{ \frac{76}{3} }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{800}{3} }[/math] [math]\displaystyle{ \frac{224}{3} }[/math] [math]\displaystyle{ 80 }[/math] [math]\displaystyle{ -\frac{256}{3} }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ -\frac{928}{3} }[/math] [math]\displaystyle{ -\frac{608}{3} }[/math] [math]\displaystyle{ \frac{1769}{15} }[/math] [math]\displaystyle{ \frac{948}{5} }[/math] [math]\displaystyle{ -\frac{13084}{45} }[/math] [math]\displaystyle{ \frac{1111}{9} }[/math] [math]\displaystyle{ -\frac{1591}{15} }[/math]

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]2 is the signature of K11a99. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 15                       1 1 13                     3   -3 11                   6 1   5 9                 8 3     -5 7               11 6       5 5             11 8         -3 3           10 11           -1 1         9 12             3 -1       5 9               -4 -3     3 9                 6 -5   1 5                   -4 -7   3                     3 -9 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=1 }[/math] [math]\displaystyle{ i=3 }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages. See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate). See/edit the Hoste-Thistlethwaite_Splice_Base (expert). Back to the top.

K11a98 K11a100