# 5 1

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 5 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 5 1 at Knotilus! An interlaced pentagram, this is known variously as the "Cinquefoil Knot", after certain herbs and shrubs of the rose family which have 5-lobed leaves and 5-petaled flowers (see e.g. [4]), as the "Pentafoil Knot" (visit Bert Jagers' pentafoil page), as the "Double Overhand Knot", as 5_1, or finally as the torus knot T(5,2).

 A kolam of a 2x3 dot array The VISA Interlink Logo [1] Version of the US bicentennial emblem A pentagonal table by Bob Mackay [2] The Utah State Parks logo As impossible object ("Penrose" pentagram) Folded ribbon which is single-sided (more complex version of Möbius Strip). Non-pentagonal shape. Pentagram of circles. Alternate pentagram of intersecting circles. 3D-looking rendition. Partial view of US bicentennial logo on a shirt seen in Lisboa [3] Non-prime knot with two 5_1 configurations on a closed loop. Knotted epitrochoid Sum of two 5_1s, Vienna, orthodox church

This sentence was last edited by Dror. Sometime later, Scott added this sentence.

### Knot presentations

 Planar diagram presentation X1627 X3849 X5,10,6,1 X7283 X9,4,10,5 Gauss code -1, 4, -2, 5, -3, 1, -4, 2, -5, 3 Dowker-Thistlethwaite code 6 8 10 2 4 Conway Notation [5]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 5, width is 2,

Braid index is 2

[{7, 2}, {1, 3}, {2, 4}, {3, 5}, {4, 6}, {5, 7}, {6, 1}]
 Knot 5_1. A graph, knot 5_1.

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 2 Super bridge index 3 Nakanishi index 1 Maximal Thurston-Bennequin number [-10][3] Hyperbolic Volume Not hyperbolic A-Polynomial See Data:5 1/A-polynomial

### Four dimensional invariants

 Smooth 4 genus ${\displaystyle 2}$ Topological 4 genus ${\displaystyle 2}$ Concordance genus ${\displaystyle {\textrm {ConcordanceGenus}}({\textrm {Knot}}(5,1))}$ Rasmussen s-Invariant -4

### Polynomial invariants

 Alexander polynomial ${\displaystyle t^{2}+t^{-2}-t-t^{-1}+1}$ Conway polynomial ${\displaystyle z^{4}+3z^{2}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{1\}}$ Determinant and Signature { 5, -4 } Jones polynomial ${\displaystyle -q^{-7}+q^{-6}-q^{-5}+q^{-4}+q^{-2}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle a^{6}\left(-z^{2}\right)-2a^{6}+a^{4}z^{4}+4a^{4}z^{2}+3a^{4}}$ Kauffman polynomial (db, data sources) ${\displaystyle a^{9}z+a^{8}z^{2}+a^{7}z^{3}-a^{7}z+a^{6}z^{4}-3a^{6}z^{2}+2a^{6}+a^{5}z^{3}-2a^{5}z+a^{4}z^{4}-4a^{4}z^{2}+3a^{4}}$ The A2 invariant ${\displaystyle -q^{22}-q^{20}-q^{18}+q^{14}+q^{12}+2q^{10}+q^{8}+q^{6}}$ The G2 invariant ${\displaystyle q^{120}-q^{100}-q^{98}-q^{92}-q^{90}-q^{88}-q^{82}-q^{80}-q^{78}-q^{72}+q^{58}+q^{56}+q^{52}+2q^{50}+q^{48}+q^{46}+q^{44}+q^{42}+2q^{40}+q^{38}+q^{34}+q^{32}+q^{30}}$

### "Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {[[10_132]], }

Same Jones Polynomial (up to mirroring, ${\displaystyle q\leftrightarrow q^{-1}}$): {[[10_132]], }

### Vassiliev invariants

 V2 and V3: (3, -5)
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 ${\displaystyle 12}$ ${\displaystyle -40}$ ${\displaystyle 72}$ ${\displaystyle 174}$ ${\displaystyle 26}$ ${\displaystyle -480}$ ${\displaystyle -{\frac {2512}{3}}}$ ${\displaystyle -{\frac {448}{3}}}$ ${\displaystyle -104}$ ${\displaystyle 288}$ ${\displaystyle 800}$ ${\displaystyle 2088}$ ${\displaystyle 312}$ ${\displaystyle {\frac {41151}{10}}}$ ${\displaystyle {\frac {2494}{15}}}$ ${\displaystyle {\frac {7634}{5}}}$ ${\displaystyle {\frac {43}{2}}}$ ${\displaystyle {\frac {1951}{10}}}$

V2,1 through V6,9 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 ${\displaystyle t^{r}q^{j}}$ are shown, along with their alternating sums ${\displaystyle \chi }$ (fixed ${\displaystyle j}$, alternation over ${\displaystyle r}$). The squares with yellow highlighting are those on the "critical diagonals", where ${\displaystyle j-2r=s+1}$ or ${\displaystyle j-2r=s-1}$, where ${\displaystyle s=}$-4 is the signature of 5 1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-5-4-3-2-10χ
-3     11
-5     11
-7   1  1
-9      0
-11 11   0
-13      0
-151     -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-5}$ ${\displaystyle i=-3}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$