K11a171

K11a170

K11a172

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a171 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1,2\}} 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a171/ThurstonBennequinNumber Hyperbolic Volume 18.6712 A-Polynomial See Data:K11a171/A-polynomial

[edit Notes for K11a171's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4} Rasmussen s-Invariant -2

[edit Notes for K11a171's four dimensional invariants]

Polynomial invariants

Alexander polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -t^4+7 t^3-21 t^2+39 t-47+39 t^{-1} -21 t^{-2} +7 t^{-3} - t^{-4} }
Conway polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^8-z^6+z^4+2 z^2+1}
2nd Alexander ideal (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}
Determinant and Signature	{ 183, 2 }
Jones polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^8+5 q^7-12 q^6+19 q^5-26 q^4+30 q^3-29 q^2+26 q-18+11 q^{-1} -5 q^{-2} + q^{-3} }
HOMFLY-PT polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^8 a^{-2} -4 z^6 a^{-2} +2 z^6 a^{-4} +z^6-5 z^4 a^{-2} +5 z^4 a^{-4} -z^4 a^{-6} +2 z^4+z^2 a^{-2} +2 z^2 a^{-4} -z^2 a^{-6} +4 a^{-2} -2 a^{-4} -1}
Kauffman polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3 z^{10} a^{-2} +3 z^{10} a^{-4} +9 z^9 a^{-1} +20 z^9 a^{-3} +11 z^9 a^{-5} +23 z^8 a^{-2} +29 z^8 a^{-4} +16 z^8 a^{-6} +10 z^8+5 a z^7-7 z^7 a^{-1} -23 z^7 a^{-3} +z^7 a^{-5} +12 z^7 a^{-7} +a^2 z^6-65 z^6 a^{-2} -71 z^6 a^{-4} -23 z^6 a^{-6} +5 z^6 a^{-8} -21 z^6-9 a z^5-15 z^5 a^{-1} -18 z^5 a^{-3} -28 z^5 a^{-5} -15 z^5 a^{-7} +z^5 a^{-9} -a^2 z^4+47 z^4 a^{-2} +47 z^4 a^{-4} +12 z^4 a^{-6} -3 z^4 a^{-8} +14 z^4+4 a z^3+14 z^3 a^{-1} +21 z^3 a^{-3} +17 z^3 a^{-5} +6 z^3 a^{-7} -6 z^2 a^{-2} -8 z^2 a^{-4} -4 z^2 a^{-6} -2 z^2-z a^{-1} -z a^{-3} -z a^{-5} -z a^{-7} -4 a^{-2} -2 a^{-4} -1}
The A2 invariant	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^8-3 q^6+3 q^4-3 q^2-1+6 q^{-2} -4 q^{-4} +8 q^{-6} -2 q^{-8} + q^{-10} + q^{-12} -6 q^{-14} +4 q^{-16} -3 q^{-18} +2 q^{-22} - q^{-24} }
The G2 invariant	$q^{46}-4q^{44}+11q^{42}-24q^{40}+36q^{38}-43q^{36}+30q^{34}+20q^{32}-100q^{30}+210q^{28}-303q^{26}+316q^{24}-204q^{22}-78q^{20}+476q^{18}-859q^{16}+1062q^{14}-921q^{12}+376q^{10}+457q^{8}-1316q^{6}+1850q^{4}-1784q^{2}+1060+97q^{-2}-1270q^{-4}+1969q^{-6}-1893q^{-8}+1078q^{-10}+148q^{-12}-1230q^{-14}+1686q^{-16}-1303q^{-18}+249q^{-20}+1016q^{-22}-1900q^{-24}+2001q^{-26}-1197q^{-28}-222q^{-30}+1715q^{-32}-2695q^{-34}+2772q^{-36}-1870q^{-38}+304q^{-40}+1366q^{-42}-2538q^{-44}+2797q^{-46}-2076q^{-48}+684q^{-50}+801q^{-52}-1800q^{-54}+1931q^{-56}-1212q^{-58}+9q^{-60}+1116q^{-62}-1648q^{-64}+1352q^{-66}-394q^{-68}-811q^{-70}+1726q^{-72}-1967q^{-74}+1468q^{-76}-447q^{-78}-674q^{-80}+1480q^{-82}-1735q^{-84}+1427q^{-86}-747q^{-88}-13q^{-90}+601q^{-92}-883q^{-94}+847q^{-96}-591q^{-98}+270q^{-100}+17q^{-102}-196q^{-104}+252q^{-106}-230q^{-108}+152q^{-110}-71q^{-112}+14q^{-114}+22q^{-116}-31q^{-118}+28q^{-120}-20q^{-122}+10q^{-124}-4q^{-126}+q^{-128}$

Further Quantum Invariants

K11a171 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["K11a171"];

In[4]:=

Alexander[K][t]

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

Out[4]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -t^4+7 t^3-21 t^2+39 t-47+39 t^{-1} -21 t^{-2} +7 t^{-3} - t^{-4} }

In[5]:=

Conway[K][z]

Out[5]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^8-z^6+z^4+2 z^2+1}

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]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 183, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^8+5 q^7-12 q^6+19 q^5-26 q^4+30 q^3-29 q^2+26 q-18+11 q^{-1} -5 q^{-2} + q^{-3} }

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^8 a^{-2} -4 z^6 a^{-2} +2 z^6 a^{-4} +z^6-5 z^4 a^{-2} +5 z^4 a^{-4} -z^4 a^{-6} +2 z^4+z^2 a^{-2} +2 z^2 a^{-4} -z^2 a^{-6} +4 a^{-2} -2 a^{-4} -1}

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3 z^{10} a^{-2} +3 z^{10} a^{-4} +9 z^9 a^{-1} +20 z^9 a^{-3} +11 z^9 a^{-5} +23 z^8 a^{-2} +29 z^8 a^{-4} +16 z^8 a^{-6} +10 z^8+5 a z^7-7 z^7 a^{-1} -23 z^7 a^{-3} +z^7 a^{-5} +12 z^7 a^{-7} +a^2 z^6-65 z^6 a^{-2} -71 z^6 a^{-4} -23 z^6 a^{-6} +5 z^6 a^{-8} -21 z^6-9 a z^5-15 z^5 a^{-1} -18 z^5 a^{-3} -28 z^5 a^{-5} -15 z^5 a^{-7} +z^5 a^{-9} -a^2 z^4+47 z^4 a^{-2} +47 z^4 a^{-4} +12 z^4 a^{-6} -3 z^4 a^{-8} +14 z^4+4 a z^3+14 z^3 a^{-1} +21 z^3 a^{-3} +17 z^3 a^{-5} +6 z^3 a^{-7} -6 z^2 a^{-2} -8 z^2 a^{-4} -4 z^2 a^{-6} -2 z^2-z a^{-1} -z a^{-3} -z a^{-5} -z a^{-7} -4 a^{-2} -2 a^{-4} -1}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q\leftrightarrow q^{-1}} ): {}

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["K11a171"];

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]=

{ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -t^4+7 t^3-21 t^2+39 t-47+39 t^{-1} -21 t^{-2} +7 t^{-3} - t^{-4} } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^8+5 q^7-12 q^6+19 q^5-26 q^4+30 q^3-29 q^2+26 q-18+11 q^{-1} -5 q^{-2} + q^{-3} } }

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]=

{}

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 8} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 16} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 32} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{172}{3}} ${\frac {20}{3}}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 128} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{544}{3}} ${\frac {64}{3}}$ $16$ ${\frac {256}{3}}$ $128$ ${\frac {1376}{3}}$ ${\frac {160}{3}}$ ${\frac {10471}{15}}$ ${\frac {1636}{15}}$ ${\frac {5404}{45}}$ ${\frac {89}{9}}$ $-{\frac {89}{15}}$

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 $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). 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 K11a171. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 17                       1 -1 15                     4   4 13                   8 1   -7 11                 11 4     7 9               15 8       -7 7             15 11         4 5           14 15           1 3         12 15             -3 1       7 15               8 -1     4 11                 -7 -3   1 7                   6 -5   4                     -4 -7 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}} $r=-1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=0$ ${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{11}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{12}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{14}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{14}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{15}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{15}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{15}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{15}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=4} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{11}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=5} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{8}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=6} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2^{4}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=7} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}

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.

K11a170 K11a172